Abstract

In optical tweezers applications, tracking a trapped particle is essential for force measurement. One of the most popular techniques for single-particle tracking is achieved by analyzing the forward and backward light pattern, scattered by the target particle trapped by a trap laser beam, of an additional probe-laser beam with different wavelength whose focus is slightly apart from the trapping center. However, the optimized focal offset has never been discussed. In this paper, we investigate the tracking range and sensitivity as a function of the focal offset between the trapping and the probe-laser beams. As a result, the optimized focal offsets are a 3.3-fold radius ahead and a 2.0-fold radius behind the trapping laser focus in the forward tracking and the backward tracking, respectively. The experimental result agrees well with a theoretical prediction using the Mie scattering theory.

© 2012 Optical Society of America

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References

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  1. L. P. Ghislain and W. W. Webb, “Scanning force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
    [CrossRef]
  2. L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
    [CrossRef]
  3. E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
    [CrossRef]
  4. M. E. J. Friese, H. Rubinsztein-Dunlop, N. R. Heckenberg, and E. W. Dearden, “Determination of the force constant of a single-beam gradient trap by measurement of backscattered light,” Appl. Opt. 35, 7112–7116 (1996).
    [CrossRef]
  5. A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
    [CrossRef]
  6. F. Gittes and C. Schmidt, “Signal and noise in micromechanical measurements,” Methods Cell Biol. 55, 129–156 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. A. Rohrbach and E. H. K. Stelzer, “Three-dimensional position detection of optically trapped dielectric particles,” J. Appl. Phys. 91, 5474–5488 (2002).
    [CrossRef]
  22. A. Rohrbach, H. Kress, and E. H. K. Stelzer, “Three-dimensional tracking of small spheres in focused laser beams: influence of the detection angular aperture,” Opt. Lett. 28, 411–413 (2003).
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    [CrossRef]

2010 (1)

2009 (1)

2007 (1)

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

2005 (1)

2004 (3)

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

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

A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
[CrossRef]

2003 (1)

2002 (2)

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

A. Rohrbach and E. H. K. Stelzer, “Trapping force, force constants, and potential depth for dielectric sphere in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
[CrossRef]

2000 (2)

B. Ovryn and S. H. Izen, “Imaging of transparent spheres through a planar interface using a high-numerical-aperture optical microscope,” J. Opt. Soc. Am. A 17, 1202–1213 (2000).
[CrossRef]

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[CrossRef]

1999 (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]

1998 (2)

F. Gittes and C. Schmidt, “Signal and noise in micromechanical measurements,” Methods Cell Biol. 55, 129–156 (1998).
[CrossRef]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

1997 (1)

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

1996 (2)

E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

M. E. J. Friese, H. Rubinsztein-Dunlop, N. R. Heckenberg, and E. W. Dearden, “Determination of the force constant of a single-beam gradient trap by measurement of backscattered light,” Appl. Opt. 35, 7112–7116 (1996).
[CrossRef]

1994 (1)

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

1993 (1)

1989 (1)

1986 (1)

Bennink, M. L.

Berg-Sørensen, K.

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

Bernet, S.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[CrossRef]

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Bouyer, P.

Buosciolo, A.

A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
[CrossRef]

Dearden, E. W.

Dulin, D.

Florin, E.

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

Florin, E. L.

E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

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]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

Flyvbjerg, H.

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

Friedrich, L.

Friese, M. E. J.

Ghislain, L. P.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

L. P. Ghislain and W. W. Webb, “Scanning force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
[CrossRef]

Gittes, F.

F. Gittes and C. Schmidt, “Signal and noise in micromechanical measurements,” Methods Cell Biol. 55, 129–156 (1998).
[CrossRef]

Heckenberg, N. R.

Hecker, N.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[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]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

Huisstede, J. H. G.

Izen, S. H.

Kozyreff, G.

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

Kress, H.

Larkin, K. G.

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Le Gall, A.

Mansuripur, M.

Neumayer, D.

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

Ovryn, B.

Perronet, K.

Pesce, G.

A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
[CrossRef]

Petrov, D.

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

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]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

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]

Ritsch-Marte, M.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[CrossRef]

Rohrbach, A.

Rubinsztein-Dunlop, H.

Sasso, A.

A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
[CrossRef]

Schmidt, C.

F. Gittes and C. Schmidt, “Signal and noise in micromechanical measurements,” Methods Cell Biol. 55, 129–156 (1998).
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Singer, W.

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[CrossRef]

Speidel, M.

Stelzer, E.

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

Stelzer, E. H. K.

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

A. Rohrbach and E. H. K. Stelzer, “Trapping force, force constants, and potential depth for dielectric sphere in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
[CrossRef]

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

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]

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

Subramaniam, V.

Switz, N. A.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

Tischer, C.

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, “Mie’s formula solution,” in Light Scattering by Small Particles (Dover, 1981), pp. 119–128.

van der Werf, K. O.

Villing, A.

Visscher, K.

Volpe, G.

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

Webb, W. W.

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

L. P. Ghislain and W. W. Webb, “Scanning force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
[CrossRef]

Westbrook, N.

Appl. Opt. (2)

Appl. Phys. (1)

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Hörber, “Photonic force microscope calibration by thermal noise analysis,” Appl. Phys. 66, S75–S78 (1998).
[CrossRef]

Appl. Phys. Lett. (1)

E. L. Florin, J. K. H. Hörber, and E. H. K. Stelzer, “High-resolution axial and lateral position sensing using two-photon excitation of fluorophores by a continuous-wave Nd:YAG laser,” Appl. Phys. Lett. 69, 446–448 (1996).
[CrossRef]

J. Appl. Phys. (2)

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

G. Volpe, G. Kozyreff, and D. Petrov, “Backscattering position detection for photonic force microscopy,” J. Appl. Phys. 102, 084701 (2007).
[CrossRef]

J. Mod. Opt. (1)

W. Singer, S. Bernet, N. Hecker, and M. Ritsch-Marte, “Three-dimensional force calibration of optical tweezers,” J. Mod. Opt. 47, 2921–2931 (2000).
[CrossRef]

J. Opt. Soc. Am. A (3)

Methods Cell Biol. (1)

F. Gittes and C. Schmidt, “Signal and noise in micromechanical measurements,” Methods Cell Biol. 55, 129–156 (1998).
[CrossRef]

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]

Opt. Commun. (1)

A. Buosciolo, G. Pesce, and A. Sasso, “New calibration method for position detector for simultaneous measurements of force constants and local viscosity in optical tweezers,” Opt. Commun. 230, 357–368 (2004).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Optik (1)

C. J. R. Sheppard and K. G. Larkin, “Vectorial pupil functions and vectorial transfer functions,” Optik 107, 79–87 (1997).

Rev. Sci. Instrum. (3)

L. P. Ghislain, N. A. Switz, and W. W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
[CrossRef]

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

A. Rohrbach, C. Tischer, D. Neumayer, E. Florin, and E. Stelzer, “Trapping and tracking a local probe with a photonic force microscope,” Rev. Sci. Instrum. 75, 2197–2210(2004).
[CrossRef]

Other (2)

H. C. van de Hulst, “Mie’s formula solution,” in Light Scattering by Small Particles (Dover, 1981), pp. 119–128.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 1983).

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Figures (6)

Fig. 1.
Fig. 1.

Schematic of optical model and coordinate geometry. OBJ and CL express the objective and the condenser, respectively. The vector bp is the position of the tracked particle from the probe laser, and the vector b is the particle displacement according to the trapping laser focus. The parameter zp is the focal offset between the trapping laser and the probe laser focuses.

Fig. 2.
Fig. 2.

Setup of single-particle tracking system. L1, L2, expander lens set; M, mirror; OBJ, objective; L3, tube lens; CL, condenser; QPD, quadrant photodiode; PZT, piezotranslation stage; CCD, camera.

Fig. 3.
Fig. 3.

Experimental result of a 0.5 μm in-radius (1.05λ) bead in the FS configuration. The intensity signal Sx is respected to the lateral displacement (bx, by) and the focal offset zp. In (a) and (b), the contour plots of the signals Sx vary with (bx, by, zp=2.2λ) and (bx, by=0, zp). (c) The relationship between Sx and bx, where (by,zp)=(0,2.2λ). In the region marked solid box, Sx varies linearly only with the lateral displacement bx. In (d), the optimization factor Wx, as a function of focal offset zp, also achieves the maximum, when zp=2.2λ.

Fig. 4.
Fig. 4.

Optimized focal offsets for different-size beads obtained from experimental data (the solid squares ▪) and theoretical simulations (the gray lines and the solid circles •).

Fig. 5.
Fig. 5.

Tracking range and signal sensitivity for different-size beads and different axes in the FS configuration. In (b), the vertical axis is the resolution defined as the reciprocal of the signal sensitivity (1/vi).

Fig. 6.
Fig. 6.

Tracking range and signal sensitivity for different-size beads and different axes in the BS configuration.

Equations (10)

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

E˜i(kx,ky)=E0exp[kin2/(k0NAOBJ)2]exp[ikz·z]Pin(kx,ky),
Pin(kin)=1k0nm[1kx2/(1+kz)kxky/(1+kz)kx],
Ein(r)=1(2π)2kink0NAOBJE˜in(kx,ky)exp[ik·r]dkxdky.
E˜s(ks)=exp[iks·bp]kz/(k0nm)(kx2+ky2)1/2k0NAOBJRin1(kin)·Es,1[Rin(kin)·ks]·exp[ikin·bp]dkxdky,
E˜s,1(ks)=E0/(k0nm)·[cosφs·S2(θs)eθs+sinφs·S1(θs)eφs],
Rin(kin)=1k0n[1kx2/(1+kz)kxky/(1+kz)kxkxky/(1+kz)1ky2/(1+kz)kykxkykz].
If(ζf,ηf)={ε0c|E˜i(ζf,ηf)+E˜s(ζf,ηf)|2kz0,(ζf2+ηf2)(k0NACL)20otherwise,
Ib(ζb,ηb)={ε0c|F1{Ri(ks)E˜s(kx,ky)}ζb,ηb|2kz0,(kx2+ky2)(k0NAOBJ)20otherwise.
[SxSySz]FS=1P0[xf0Ifdxfdyfxf0Ifdxfdyfyf0Ifdxfdyfyf0IfdxfdyfIfdxfdyf],
[SxSySz]BS=1P0[xb0Ibdxbdybxb0Ibdxbdybyb0Ibdxbdybyb0IbdxbdybIbdxbdyb].

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