Abstract

Interparticle interaction energies and other useful physical characteristics can be extracted from the statistical properties of the motion of particles confined by an optical line trap. In practice, however, the potential energy landscape, U(x), imposed by the line provides an extra, and in general unknown, influence on particle dynamics. We describe a new class of line traps in which both the optical gradient and scattering forces acting on a trapped particle are designed to be linear functions of the line coordinate and in which their magnitude can be counterbalanced to yield a flat U(x). These traps are formed using approximate solutions to general relations concerning non-conservative optical forces that have been the subject of recent investigations [Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, Phys. Rev. Lett. 100, 013602-4 (2008).]. We implement the lines using holographic optical trapping and measure the forces acting on silica microspheres, demonstrating the tunability of the confining potential energy landscape. Furthermore, we show that our approach efficiently directs available laser power to the trap, in contrast to other methods.

© 2008 Optical Society of America

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  1. A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
    [CrossRef]
  2. D. G. Grier, "A Revolution in Optical Manipulation," Nature 424, 810-816 (2003).
    [CrossRef]
  3. E. M. Furst, "Applications of laser tweezers in complex fluid rheology," Curr. Opin. Colloid Interface Sci. 10, 79-86 (2005).
    [CrossRef]
  4. J. C. Crocker and D. G. Grier, "Microscopic measurement of the pair interaction potential of charge-stabilized colloid," Phys. Rev. Lett. 73, 352-355 (1994).
    [CrossRef]
  5. J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
    [CrossRef]
  6. K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
    [CrossRef]
  7. P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
    [CrossRef]
  8. Y. Roichman and D. G. Grier, "Projecting Extended Optical Traps With Shape-Phase Holography," Opt. Lett. 31, 1675-1677 (2006).
    [CrossRef]
  9. T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
    [CrossRef]
  10. R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
    [CrossRef]
  11. Optical forces are, in general, non-conservative (see Ref. [16] and references therein) and so measurements of forces reveal pseudopotentials rather than true potential energy functions. In one dimension, however, any force that depends only on position is necessarily conservative since its integral is uniquely determined. The experiments described here involve only one-dimensional force characterizations, and so determine an effective potential U(x) corresponding to the x-axis components of forces; they do not determine the 3D pseudopotential.
  12. J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
    [CrossRef]
  13. 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]
  14. G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. J. Laczik, "Assembly of 3-Dimensional Structures using programmable Holographic Optical Tweezers," Opt. Express 12, 5475-5480 (2004).
    [CrossRef]
  15. A. J. DeWeerd and S. E. Hill, "The Dizzying Depths of the Cylindrical Mirror," Phys. Teach. 43, 90-92 (2005).
    [CrossRef]
  16. Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
    [CrossRef]
  17. The CCD pixel intensity was verified to be a linear function of the applied laser power, with a coefficient of determination ("R2") of 0.9992. The intensity profiles along x are measured along the image row of greatest intensity, averaged over adjacent rows spanning ±0.3μm in y.
  18. J. C. Crocker and D. G. Grier, "Methods of Digital Video Microscopy for Colloidal Studies," J. Coll. Interf. Sci. 179, 298-310 (1996).
    [CrossRef]
  19. S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
    [CrossRef]
  20. A total of approximately ten thousand �?x values were recorded for each line. We find no apparent variation of s2x i with position or with �?m; its value yields a diffusion coefficient D = 0.068±0.006 μm2/s. In the F(x) plot of Fig. 3(b) (inset), the mean value of F is subtracted; this position-independent force is likely due to convective flow in the chamber or ravitational forces caused by substrate tilt. This offset is irrelevant to the determination of the slope, B.
  21. The value of B for any line trap was determined by a linear fit of all the �?x vs. x, to avoid artefacts related to the binning of data. As follows from the discussion in the main text, B is equal to the slope of this �?x vs. x fit times 2kBT/s2, where s2 is the position-independent mean of s2xi.
  22. M. Gu, Advanced Optical Imaging Theory (Springer, Berlin, 2000).
  23. M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
    [CrossRef]

2008

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

2007

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef]

2006

2005

M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
[CrossRef]

A. J. DeWeerd and S. E. Hill, "The Dizzying Depths of the Cylindrical Mirror," Phys. Teach. 43, 90-92 (2005).
[CrossRef]

E. M. Furst, "Applications of laser tweezers in complex fluid rheology," Curr. Opin. Colloid Interface Sci. 10, 79-86 (2005).
[CrossRef]

P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
[CrossRef]

2004

T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
[CrossRef]

G. Sinclair, P. Jordan, J. Courtial, M. Padgett, J. Cooper, and Z. J. Laczik, "Assembly of 3-Dimensional Structures using programmable Holographic Optical Tweezers," Opt. Express 12, 5475-5480 (2004).
[CrossRef]

2003

D. G. Grier, "A Revolution in Optical Manipulation," Nature 424, 810-816 (2003).
[CrossRef]

2002

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

1999

1998

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

1997

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
[CrossRef]

1996

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

1994

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

K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
[CrossRef]

Amato-Grill, J.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

Ashkin, A.

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
[CrossRef]

Biancaniello, P. L.

P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
[CrossRef]

Block, S. M.

K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
[CrossRef]

Bustamante, C.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Cheong, F.-C.

T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
[CrossRef]

Cooper, J.

Courtial, J.

Crocker, J. C.

P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
[CrossRef]

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

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

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

Curtis, J. E.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

DeWeerd, A. J.

A. J. DeWeerd and S. E. Hill, "The Dizzying Depths of the Cylindrical Mirror," Phys. Teach. 43, 90-92 (2005).
[CrossRef]

Dufresne, E. R.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef]

M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
[CrossRef]

Dumont, S.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Furst, E. M.

E. M. Furst, "Applications of laser tweezers in complex fluid rheology," Curr. Opin. Colloid Interface Sci. 10, 79-86 (2005).
[CrossRef]

Germain, V.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef]

Grier, D. G.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

Y. Roichman and D. G. Grier, "Projecting Extended Optical Traps With Shape-Phase Holography," Opt. Lett. 31, 1675-1677 (2006).
[CrossRef]

D. G. Grier, "A Revolution in Optical Manipulation," Nature 424, 810-816 (2003).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

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

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

Haist, T.

Hill, S. E.

A. J. DeWeerd and S. E. Hill, "The Dizzying Depths of the Cylindrical Mirror," Phys. Teach. 43, 90-92 (2005).
[CrossRef]

Hsu, M. F.

M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
[CrossRef]

Jordan, P.

Kim, A. J.

P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
[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]

Laczik, Z. J.

Liphardt, J.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Lubensky, T. C.

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

Padgett, M.

Reicherter, M.

Roichman, Y.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

Y. Roichman and D. G. Grier, "Projecting Extended Optical Traps With Shape-Phase Holography," Opt. Lett. 31, 1675-1677 (2006).
[CrossRef]

Sainis, S. K.

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef]

Sinclair, G.

Smith, S. B.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Sow, C.-H.

T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
[CrossRef]

Sun, B.

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

Svoboda, K.

K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
[CrossRef]

Tinoco, I.

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Tiziani, H. J.

Verma, R.

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

Wagemann, E. U.

Weitz, D. A.

M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
[CrossRef]

Yodh, A. G.

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

Yu, T.

T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
[CrossRef]

Annu. Rev. Biophys. Biomol. Struct.

K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
[CrossRef]

Curr. Opin. Colloid Interface Sci.

E. M. Furst, "Applications of laser tweezers in complex fluid rheology," Curr. Opin. Colloid Interface Sci. 10, 79-86 (2005).
[CrossRef]

J. Coll. Interf. Sci.

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

Langmuir

M. F. Hsu, E. R. Dufresne, and D. A. Weitz, "Charge stabilization in nonpolar solvents," Langmuir 21, 4881-4887 (2005).
[CrossRef]

Nanotechnology

T. Yu, F.-C. Cheong, and C.-H. Sow, "The manipulation and assembly of CuO nanorods with line optical tweezers," Nanotechnology 15, 1732-1736 (2004).
[CrossRef]

Nature

D. G. Grier, "A Revolution in Optical Manipulation," Nature 424, 810-816 (2003).
[CrossRef]

Opt. Commun.

J. E. Curtis, B. A. Koss, and D. G. Grier, "Dynamic holographic optical tweezers," Opt. Commun. 207, 169-175 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

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

P. L. Biancaniello, A. J. Kim, and J. C. Crocker, "Colloidal interactions and self-assembly using DNA hybridization," Phys. Rev. Lett. 94, 058302 (2005).
[CrossRef]

Y. Roichman, B. Sun, Y. Roichman, J. Amato-Grill, and D. G. Grier, "Optical Forces Arising from Phase Gradients," Phys. Rev. Lett. 100, 013602-4 (2008).
[CrossRef]

R. Verma, J. C. Crocker, T. C. Lubensky, and A. G. Yodh, "Entropic colloidal interactions in concentrated DNA solutions," Phys. Rev. Lett. 81, 4004-4007 (1998).
[CrossRef]

S. K. Sainis, V. Germain, and E. R. Dufresne, "Statistics of particle trajectories at short time intervals reveal fN-scale colloidal forces," Phys. Rev. Lett. 99, 018303 (2007).
[CrossRef]

Phys. Teach.

A. J. DeWeerd and S. E. Hill, "The Dizzying Depths of the Cylindrical Mirror," Phys. Teach. 43, 90-92 (2005).
[CrossRef]

Proc. Natl. Acad. Sci. USA

A. Ashkin, "Optical trapping and manipulation of neutral particles using lasers," Proc. Natl. Acad. Sci. USA 94, 4853-4860 (1997).
[CrossRef]

Science

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco, Jr., and C. Bustamante, "Equilibrium information from nonequilibrium measurements in an experimental test of Jarzynski�??s equality," Science 296, 1832-1835 (2002).
[CrossRef]

Other

The CCD pixel intensity was verified to be a linear function of the applied laser power, with a coefficient of determination ("R2") of 0.9992. The intensity profiles along x are measured along the image row of greatest intensity, averaged over adjacent rows spanning ±0.3μm in y.

A total of approximately ten thousand �?x values were recorded for each line. We find no apparent variation of s2x i with position or with �?m; its value yields a diffusion coefficient D = 0.068±0.006 μm2/s. In the F(x) plot of Fig. 3(b) (inset), the mean value of F is subtracted; this position-independent force is likely due to convective flow in the chamber or ravitational forces caused by substrate tilt. This offset is irrelevant to the determination of the slope, B.

The value of B for any line trap was determined by a linear fit of all the �?x vs. x, to avoid artefacts related to the binning of data. As follows from the discussion in the main text, B is equal to the slope of this �?x vs. x fit times 2kBT/s2, where s2 is the position-independent mean of s2xi.

M. Gu, Advanced Optical Imaging Theory (Springer, Berlin, 2000).

Optical forces are, in general, non-conservative (see Ref. [16] and references therein) and so measurements of forces reveal pseudopotentials rather than true potential energy functions. In one dimension, however, any force that depends only on position is necessarily conservative since its integral is uniquely determined. The experiments described here involve only one-dimensional force characterizations, and so determine an effective potential U(x) corresponding to the x-axis components of forces; they do not determine the 3D pseudopotential.

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