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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References

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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] [PubMed]
  2. D. G. Grier, "A Revolution in Optical Manipulation," Nature 424, 810-816 (2003).
    [CrossRef] [PubMed]
  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] [PubMed]
  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] [PubMed]
  6. K. Svoboda and S. M. Block, "Biological applications of optical forces" Annu. Rev. Biophys. Biomol. Struct. 23, 247-285 (1994).
    [CrossRef] [PubMed]
  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] [PubMed]
  8. Y. Roichman and D. G. Grier, "Projecting Extended Optical Traps With Shape-Phase Holography," Opt. Lett. 31, 1675-1677 (2006).
    [CrossRef] [PubMed]
  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] [PubMed]
  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] [PubMed]
  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] [PubMed]
  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] [PubMed]

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] [PubMed]

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] [PubMed]

2006

2005

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

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

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] [PubMed]

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

2004

2003

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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

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] [PubMed]

Ashkin, A.

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

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] [PubMed]

Block, S. M.

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

Svoboda, K.

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

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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.

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] [PubMed]

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] [PubMed]

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] [PubMed]

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]

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] [PubMed]

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] [PubMed]

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] [PubMed]

Other

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.

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).

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.

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Figures (3)

Fig. 1.
Fig. 1.

(a) Simplified schematic of the setup highlighting coordinate axes and focal planes. Thick arrow: a beam deflected from the optical axis (z). Thin arrow: an undeflected beam. For clarity, angles are greatly exaggerated and refraction at the lenses is only approximately indicated. (b) Illustration of the mean radiation force (darker gray arrow) and its nonzero x-component (lighter gray arrow) for a particle located at a position in the line trap (red line) away from the optical axis.

Fig. 2.
Fig. 2.

(a) Phase profiles ϕ(u) for various values of σ with parameters L=60µm, f=2.57 mm, λ=0.655µm, and u max=3.6 mm. (b) Measured intensity profiles, I(x), of Gaussian line traps, together with Gaussian fits. Inset: Images of two of the lines. (c) Measured (σm ) and intended (σi ) widths. A linear fit of σ -2 m vs. σ -2 i has slope 0.99 ± 0.3. Inset: Integrated line intensity as a function of σ

Fig. 3.
Fig. 3.

(a) Representative trajectories of radius a=1.6µm silica microspheres initially at the center (x=0) of line traps with intensities I(x)∝exp(-x 2/(2σ 2)). The σi values are listed first, with the measured σm in parentheses. (b) The dependence of B, the measured slope of F vs. x, on the Gaussian width σm . B crosses zero at σm =8.4±0.7µm, revealing the “flat” line trap for which radiation and gradient forces balance. Inset: The measured position-dependent force, F(x), for σi =∞.

Equations (11)

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

ϕ ( u ) = π Lu 2 ( 2 λ fu max ) .
x I ( x ) A d I d x = 0 ,
I ( x ) = C exp ( x 2 ( 2 σ 2 ) ) ,
d u d ϕ′ ~ exp ( ( λf 2 π ) 2 ( ϕ′ ) 2 2 σ 2 ) .
ϕ′ ( u ) = ( 2 π 2 σ ( λ f ) ) erfinv ( u D ) .
ϕ′ ( u = ± u max ) = 2 π λ f ( ± L 2 ) = ± π L λ f ,
ϕ′ ( u ) = ( 2 2 π λ f σ ) erfinv ( u u max erf ( L 2 2 σ ) ) .
E ~ ( x ) = 1 f λ E ( u ) exp ( j 2 π f λ xu ) du .
ϕ ( x ) = 2 π u max f λ L x 2 .
ϕ ( x ) = 2 π u max f λ L x 2 [ 1 4 b a 3 ( π f λ ) 2 x 2 ] ,
d ϕ d x I ( x ) A d I d x = 0 .

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