Abstract

Holographic optical tweezers (HOTs) enable the manipulation of multiple traps independently in three dimensions in real time. Application of this technique to force measurements requires calibration of trap stiffness and its position dependence. Here, we determine the trap stiffness of HOTs as they are steered in two dimensions. To do this, we trap a single particle in a multiple-trap configuration and analyze the power spectrum of the laser deflection on a position-sensitive photodiode. With this method, the relative trap strengths can be determined independent of exact particle size, and high stiffnesses can be probed because of the high bandwidth of the photodiode. We find a trap stiffness for each of three HOT traps of κ ~26 pN/µm per 100 mW of laser power. Importantly, we find that this stiffness remains constant within ±4% over 20µm displacements of a trap. We also investigate the minimum step size achievable when steering a trap with HOTs, and find that traps can be stepped and detected within ~2 nm in our instrument, although there is an underlying position modulation of the traps of comparable scale that arises from SLM addressing. The independence of trap stiffness on steering angle over wide ranges and the nanometer positioning accuracy of HOTs demonstrate the applicability of this technique to quantitative study of force response of extended biomaterials such as cells or elastomeric protein networks.

©2008 Optical Society of America

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    [Crossref] [PubMed]
  3. W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
    [Crossref] [PubMed]
  4. C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
    [Crossref] [PubMed]
  5. G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
    [Crossref] [PubMed]
  6. K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
    [Crossref] [PubMed]
  7. D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
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  8. Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
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    [Crossref]
  10. M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
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  22. E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
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    [Crossref]
  27. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [Crossref]
  28. M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
    [Crossref] [PubMed]
  29. S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
    [Crossref]
  30. 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, 1074–1085 (1998).
    [Crossref] [PubMed]
  31. S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
    [Crossref]
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    [Crossref] [PubMed]
  37. M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Lett. 33, 599–601 (2008).
    [Crossref] [PubMed]
  38. A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
    [Crossref] [PubMed]
  39. 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, 18275–18282 (2007).
    [Crossref] [PubMed]

2008 (4)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Lett. 33, 599–601 (2008).
[Crossref] [PubMed]

F. Belloni, S. Monneret, F. Monduc, and M. Scordia, “Multiple holographic optical tweezers parallel calibration with optical potential well characterization,” Opt. Express 16, 9011–9020 (2008).
[Crossref] [PubMed]

2007 (7)

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[Crossref] [PubMed]

E. Eriksson, S. Keen, J. Leach, M. Goksör, and M. J. Padgett, “The effect of external forces on discrete motion within holographic optical tweezers,” Opt. Express 15, 18268–18274 (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, 18275–18282 (2007).
[Crossref] [PubMed]

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
[Crossref]

W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
[Crossref] [PubMed]

Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
[Crossref]

2006 (3)

2005 (4)

2004 (5)

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

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

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

2003 (2)

E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
[Crossref]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

2002 (2)

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31–35 (2002).

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

2001 (2)

2000 (1)

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

1999 (2)

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[Crossref]

1998 (2)

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, 1074–1085 (1998).
[Crossref] [PubMed]

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[Crossref]

1996 (1)

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

1993 (1)

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[Crossref] [PubMed]

1986 (1)

1969 (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[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, 1074–1085 (1998).
[Crossref] [PubMed]

Andreasson, J. O.

Ashkin, A.

Bechhoefer, J.

Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
[Crossref]

Belloni, F.

Berg-Sørensen, K.

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

Bjorkholm, J. E.

Block, S. M.

M. T. Valentine, N. R. Guydosh, B. Gutiérrez-Medina, A. N. Fehr, J. O. Andreasson, and S. M. Block, “Precision steering of an optical trap by electro-optic deflection,” Opt. Lett. 33, 599–601 (2008).
[Crossref] [PubMed]

W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
[Crossref] [PubMed]

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

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

Boyle, M.

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

Brakenhoff, G. J.

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[Crossref] [PubMed]

Burnham, D. R.

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

Carter, B. C.

B. C. Carter, G. T. Shubeita, and S. P. Gross, “Tracking single particles: a user-friendly quantitative evaluation,” Phys. Biol. 2, 60–72 (2005).
[Crossref] [PubMed]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

Chu, S.

Clark, R. L.

Cole, D. G.

Cooper, J.

K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006).
[Crossref] [PubMed]

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

Courtial, J.

G. C. Spalding, J. Courtial, and R. Di Leonardo , “Holographic optical tweezers,” in Structured Light and Its ApplicationsD. L. Andrews, ed. (Academic Press, 2008) pp. 139–168.

Crossland, W. A.

Curtis, J. E.

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, 1074–1085 (1998).
[Crossref] [PubMed]

Deng, Y.

Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
[Crossref]

Dholakia, K.

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31–35 (2002).

Di Leonardo, R.

Dogterom, M.

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

Dufresne, E. R.

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[Crossref]

Dziedzic, J. M.

Eriksson, E.

Fehr, A. N.

Florin, E.-L.

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

Flyvbjerg, H.

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

Forde, N. R.

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

Forde, N.R.

Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
[Crossref]

Friederich, F.

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

Gallet, F.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

Gibson, G.

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006).
[Crossref] [PubMed]

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, 1074–1085 (1998).
[Crossref] [PubMed]

Goksör, M.

Greenleaf, W. J.

W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
[Crossref] [PubMed]

Grier, D. G.

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, 18275–18282 (2007).
[Crossref] [PubMed]

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
[Crossref] [PubMed]

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[Crossref]

Gross, S. P.

B. C. Carter, G. T. Shubeita, and S. P. Gross, “Tracking single particles: a user-friendly quantitative evaluation,” Phys. Biol. 2, 60–72 (2005).
[Crossref] [PubMed]

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

Gutiérrez-Medina, B.

Guydosh, N. R.

Haist, T.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[Crossref]

Hällstig, E.

E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
[Crossref]

Harriman, J.

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W (2006).
[Crossref]

Hénon, S.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

Hermerschmidt, A.

S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
[Crossref]

Hirsch, P. M.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Hörber, J. K. H.

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

Ianni, F.

Izhaky, D.

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

Jordan, J. A.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Jordan, P.

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

Keen, S.

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

E. Eriksson, S. Keen, J. Leach, M. Goksör, and M. J. Padgett, “The effect of external forces on discrete motion within holographic optical tweezers,” Opt. Express 15, 18268–18274 (2007).
[Crossref] [PubMed]

Kim, S.-H.

Kling, R.

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

Klug, U.

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

Koss, B. A.

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

Krol, J. J.

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[Crossref] [PubMed]

Krüger, S.

S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
[Crossref]

Ladavac, K.

Leach, J.

E. Eriksson, S. Keen, J. Leach, M. Goksör, and M. J. Padgett, “The effect of external forces on discrete motion within holographic optical tweezers,” Opt. Express 15, 18268–18274 (2007).
[Crossref] [PubMed]

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006).
[Crossref] [PubMed]

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

Lee, S.-H.

Lenormand, G.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

Lesem, L. B.

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

Liesener, J.

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

Lindgren, M.

E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
[Crossref]

MacDonald, M.

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31–35 (2002).

Manolis, I. G.

McGloin, D.

Mears, R. J.

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

Monduc, F.

Monneret, S.

Neuman, K. C.

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

Osten, S.

S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
[Crossref]

Ostendorf, A.

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

Padgett, M.

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

Padgett, M. J.

Polin, M.

Pralle, A.

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

Prummer, M.

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

Redmond, M. M.

Reicherter, M.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[Crossref]

Richert, A.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

Robertson, B.

Roichman, Y.

Ruocco, G.

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, 1074–1085 (1998).
[Crossref] [PubMed]

Schmitz, C. H. J.

Scordia, M.

Serati, S.

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W (2006).
[Crossref]

Shubeita, G. T.

B. C. Carter, G. T. Shubeita, and S. P. Gross, “Tracking single particles: a user-friendly quantitative evaluation,” Phys. Biol. 2, 60–72 (2005).
[Crossref] [PubMed]

Siméon, J.

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

Sinclair, G.

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

Sjöqvist, L.

E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
[Crossref]

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

Spalding, G.

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31–35 (2002).

Spalding, G. C.

G. C. Spalding, J. Courtial, and R. Di Leonardo , “Holographic optical tweezers,” in Structured Light and Its ApplicationsD. L. Andrews, ed. (Academic Press, 2008) pp. 139–168.

Spatz, J. P.

Stelzer, E. H. K.

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

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, 1074–1085 (1998).
[Crossref] [PubMed]

Tan, K. L.

Tiziani, H. J.

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

M. Reicherter, T. Haist, E. U. Wagemann, and H. J. Tiziani, “Optical particle trapping with computer-generated holograms written on a liquid-crystal display,” Opt. Lett. 24, 608–610 (1999).
[Crossref]

Valentine, M. T.

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, 18275–18282 (2007).
[Crossref] [PubMed]

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

van der Horst, A.

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

van Oostrum, P.

Visscher, K.

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

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[Crossref] [PubMed]

Vossen, D. L. J.

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

Wagemann, E. U.

Warr, S. T.

Wilkinson, T. D.

Woodside, M. T.

W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
[Crossref] [PubMed]

Wulff, K. D.

Yang, S.-M.

Yi, G.-R.

Annu. Rev. Biochem. (2)

C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004).
[Crossref] [PubMed]

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem.  77, 205–228 (2008).
[Crossref] [PubMed]

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

W. J. Greenleaf, M. T. Woodside, and S. M. Block, “High-resolution, single-molecule measurements of biomolecular motion,” Annu. Rev. Biophys. Biomol. Struct. 36, 171–190 (2007).
[Crossref] [PubMed]

Biophys. J. (2)

G. Lenormand, S. Hénon, A. Richert, J. Siméon, and F. Gallet, “Direct measurement of the area expansion and shear moduli of the human red blood cell membrane skeleton,” Biophys. J. 81, 43–56 (2001).
[Crossref] [PubMed]

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, 1074–1085 (1998).
[Crossref] [PubMed]

Cytometry (1)

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by multiple optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[Crossref] [PubMed]

IBM J. Res. Dev. (1)

L. B. Lesem, P. M. Hirsch, and J. A. Jordan, “The Kinoform: A New Wavefront Reconstruction Device,” IBM J. Res. Dev. 13, 150–155 (1969).
[Crossref]

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

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

J. Mod. Opt. (1)

G. Sinclair, P. Jordan, J. Leach, M. J. Padgett, and J. Cooper, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt.  51, 409–414 (2004).
[Crossref]

J. Opt. A: Pure Appl. Opt. (2)

S. Keen, J. Leach, G. Gibson, and M. Padgett, “Comparison of a high-speed camera and a quadrant detector for measuring displacements in optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S264–S266 (2007).
[Crossref]

Y. Deng, J. Bechhoefer, and N.R. Forde, “Brownian motion in a modulated optical trap,” J. Opt. A: Pure Appl. Opt. 9, S256–S263 (2007).
[Crossref]

J. Opt. Soc. Am. A (1)

Microsc. Res. Tech. (1)

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

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[Crossref] [PubMed]

Opt. Commun. (2)

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

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[Crossref]

Opt. Eng. (1)

E. Hällstig, L. Sjöqvist, and M. Lindgren, “Intensity variations using a quantized spatial light modulator for nonmechanical beam steering,” Opt. Eng. 42, 613–619 (2003).
[Crossref]

Opt. Express (8)

F. Belloni, S. Monneret, F. Monduc, and M. Scordia, “Multiple holographic optical tweezers parallel calibration with optical potential well characterization,” Opt. Express 16, 9011–9020 (2008).
[Crossref] [PubMed]

R. Di Leonardo, F. Ianni, and G. Ruocco, “Computer generation of optimal holograms for optical trap arrays,” Opt. Express 15, 1913–1922 (2007).
[Crossref] [PubMed]

D. R. Burnham and D. McGloin, “Holographic optical trapping of aerosol droplets,” Opt. Express 14, 4175–4181 (2006).
[Crossref] [PubMed]

M. Polin, K. Ladavac, S.-H. Lee, Y. Roichman, and D. G. Grier, “Optimized holographic optical traps,” Opt. Express 13, 5831–5845 (2005).
[Crossref] [PubMed]

C. H. J. Schmitz, J. P. Spatz, and J. E. Curtis, “High-precision steering of multiple holographic optical traps,” Opt. Express 13, 8678–8685 (2005).
[Crossref] [PubMed]

E. Eriksson, S. Keen, J. Leach, M. Goksör, and M. J. Padgett, “The effect of external forces on discrete motion within holographic optical tweezers,” Opt. Express 15, 18268–18274 (2007).
[Crossref] [PubMed]

K. D. Wulff, D. G. Cole, R. L. Clark, R. Di Leonardo, J. Leach, J. Cooper, G. Gibson, and M. J. Padgett, “Aberration correction in holographic optical tweezers,” Opt. Express 14, 4169–4174 (2006).
[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, 18275–18282 (2007).
[Crossref] [PubMed]

Opt. Lett. (4)

Phys. Biol. (1)

B. C. Carter, G. T. Shubeita, and S. P. Gross, “Tracking single particles: a user-friendly quantitative evaluation,” Phys. Biol. 2, 60–72 (2005).
[Crossref] [PubMed]

Phys. World (1)

K. Dholakia, G. Spalding, and M. MacDonald, “Optical tweezers: the next generation,” Phys. World 15, 31–35 (2002).

Proc. SPIE (3)

S. Osten, S. Krüger, and A. Hermerschmidt, “New HDTV (1920×1080) phase-only SLM,” Proc. SPIE 6487, 64870X (2007).
[Crossref]

U. Klug, M. Boyle, F. Friederich, R. Kling, and A. Ostendorf, “Laser beam shaping for micromaterial processing using a liquid crystal display,” Proc. SPIE 6882, 688207 (2008).
[Crossref]

S. Serati and J. Harriman, “Spatial light modulator considerations for beam control in optical manipulation applications,” Proc. SPIE 6326, 63262W (2006).
[Crossref]

Rev. Sci. Instrum. (4)

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

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

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[Crossref]

D. L. J. Vossen, A. van der Horst, M. Dogterom, and A. van Blaaderen, “Optical tweezers and confocal microscopy for simultaneous three-dimensional manipulation and imaging in concentrated colloidal dispersions,” Rev. Sci. Instrum. 75, 2960–2970 (2004).
[Crossref]

Other (1)

G. C. Spalding, J. Courtial, and R. Di Leonardo , “Holographic optical tweezers,” in Structured Light and Its ApplicationsD. L. Andrews, ed. (Academic Press, 2008) pp. 139–168.

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Figures (12)

Fig. 1.
Fig. 1. Schematic depiction of SLM beam steering. A laser beam is reflected off the SLM. A phase gradient proportional to h/l steers the beam by an angle α. Lenses L1 and L2 (in 4f-configuration) image the SLM onto the back focal plane (BFP) of the objective lens with magnification m, the ratio of the focal lengths of lenses L2 and L1. The unmodulated light (0th order) is focused on the optical axis. The modulated beam (1st order) enters the objective under an angle β=α/m. In the front focal plane (FFP) this beam is focused at a position displaced d from the zero-order peak. The dependence of d on parameters of the optical system is given by Eq. (3). This schematic is not to scale, and the phase of the laser is modulated over the entire width l of our SLM.

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Fig. 2.
Fig. 2. Theoretical SLM efficiency η due to pixelation and quantization of the phase levels (triangles, Eq. (4)); lines are a guide to the eye. Here, the maximum number of phase levels is taken to be qmax =256, as available on our SLM. For qmax =192 the values are the same (within 0.01%). The normalized angle αnorm =α/αmax , see text. Inset: expanded view of η as a function of αnorm . An angle of αnorm =0.2 corresponds to 10 pixels per 2π of phase modulation or, in our setup, d=39.9 µm.

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Fig. 3.
Fig. 3. Schematic of the holographic tweezers setup. A spatial light modulator (SLM) modulates the wavefront of an infrared laser beam. A high-numerical aperture objective focuses the light, and multiple optical traps can be created. A second identical objective captures the light, which is imaged onto a position-sensitive diode (PSD) for particle position detection. Particle imaging uses counterpropagating visible light, with the image directed to two cameras. See text for details.

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Fig. 4.
Fig. 4. Example of a measured power spectral density curve (black diamonds), which shows the expected Lorentzian behavior, along with peaks due to the intrinsic SLM address rate. These peaks are deleted prior to analysis, and a Lorentzian (red line) is fit to the spectrum from 40–2500 Hz (grey diamonds) to obtain the corner frequency fc and thereby the trap stiffness κ (Eqs. (5) and (6)). Here, fc =407 Hz and κ=51 pN/µm.

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Fig. 5.
Fig. 5. Displacement in x (a) and intensity (b) as a function of time for a single first-order beam at an angle of αnorm=0.10 in x. These were recorded on our optical table using a PSD by passing only this first-order beam through a small aperture (1 mm diameter) at the conjugate trapping plane between lens L1 and mirror M2. Shown are the data recorded for 22:6 (top) and 5:5 (middle) SLM settings, and power spectra (bottom) for both settings. Laser power was 286 mW.

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Fig. 6.
Fig. 6. (a) Laser deflection as a function of time, measured using the PSD, for 22:6 (top) and 5:5 (middle) SLM settings, and corresponding power spectra (bottom, average over 1000 spectra). A single particle was trapped in one of three HOT traps in the same configuration as in Fig. 9(a) with x=-9.9µm. (b) Position as a function of time for the same particle in the same configuration as in (a), determined from high-speed camera images (frame rate 2500 Hz), for 22:6 (top) and 5:5 (middle) SLM settings, and corresponding power spectra (bottom, average over 20 spectra). The laser power was 285 mW, measured before the focusing objective lens.

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Fig. 7.
Fig. 7. (a) Trapping configuration used for several of our experiments, imaged by back-reflection of the laser foci from the coverslip. The brightest spots are three traps #1, #2, and #3 in a triangular configuration. Also visible are the zero-order spot and ghost traps due to higher-order interference. (b) Microscope image of three 2.1-µm-diameter particles trapped in the three HOT traps shown in (a). Scale bars in both images represent 5 µm.

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Fig. 8.
Fig. 8. Trap stiffness and relative difference with respect to the mean stiffness (κrel =(κ-〈κ〉)/〈κ〉) for several positions in a triangular configuration of traps (Fig. 7), for x (solid triangles) and y (open triangles). The laser power was 188 mW, measured before the focusing objective lens. (a-b) The stiffness of trap #1, while trap #2 was displaced. (c-d) Stiffness of trap #2, for the same displacements of trap #2 as in (a-b). (e-f) Stiffness of trap #2, while both trap #1 and trap #2 were displaced. (g-h) Stiffness of trap #2, while all traps were kept stationary. See text for details. Insets: position of the traps (×) relative to each other and the range of motion of the displaced trap(s). The circle (⊗) indicates which trap is probed with the trapped particle.

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Fig. 9.
Fig. 9. Trap stiffness and relative difference with respect to the mean stiffness for x (solid triangles) and y (open triangles). Three traps were created of which one, holding the particle, was moved. (a-b) Trap moved in x-direction at y=13.2 µm; laser power was 286 mW. (c-d) Trap moved in y at x=-9.9 µm; laser power was 188 mW. The x and y positions are with respect to the zero-order spot. Insets: position of the traps (×) relative to each other and the range of motion of the displaced trap, with the circle (⊗) indicating the probed trap.

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Fig. 10.
Fig. 10. Trap stiffnesses κx (a) and κy (b) for 5:5 (open diamonds) and 22:6 (solid diamonds) SLM settings. Three traps were created of which one, holding the particle, was moved. Trap configuration was the same as in Fig. 9(a); laser power was 286 mW. The x positions are with respect to the zero-order spot.

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Fig. 11.
Fig. 11. Trap stiffnesses κx and κy as a function of laser power, for trap #2 and for the zero-order spot (see Fig. 7). Laser power is measured before entering the objective and ranges from 100 to 336 mW (#2) or to 286 mW (zeroth order). Linear fits are forced through zero.

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Fig. 12.
Fig. 12. (a) The x and y positions for trapped particles #1 (grey) and #2 (black), and the smoothed curve for x of #2 (white line). (Data are offset to fit in one graph and show positions from 68575 images recorded at 1250 fps.) (b) The x position of #1 corrected for drift using #2 (see text). The black lines show the average position over the indicated range. (c) Step size |Δx| as a function of applied difference in phase shift Δϕ, for the first series (solid triangles) and for the second (open triangles). Inset: enlarged view. Linear fit (red line). (d) Difference between |Δx| and the linear fit in (c). (For display purposes, (a) and the main graph in (b) only show 1 in every 5 data points.)

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Equations (7)

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α ≈ h l = m d f obj .
h = ϕ 2 π λ .
d = 1 m L M λ l ϕ 2 π .
η = sin c 2 ( 1 ⁄ q ) × sin c 2 ( 1 ⁄ Λ ) sin c 2 ( 1 ⁄ lcm [ q , Λ ] ) ,
S x = k B T 2 π 2 γ ( f c 2 + f 2 ) ,
κ = 2 π γ f c .
d α norm = λ f obj 2 m l pix ,

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