Abstract

A single-beam gradient trap could potentially be used to hold a stylus for scanning force microscopy. With a view to development of this technique, we modeled the optical trap as a harmonic oscillator and therefore characterized it by its force constant. We measured force constants and resonant frequencies for 1–4-μm-diameter polystyrene spheres in a single-beam gradient trap using measurements of backscattered light. Force constants were determined with both Gaussian and doughnut laser modes, with powers of 3 and 1 mW, respectively. Typical values for spring constants were measured to be between 10−6 and 4 × 10−6 N/m. The resonant frequencies of trapped particles were measured to be between 1 and 10 kHz, and the rms amplitudes of oscillations were estimated to be around 40 nm. Our results confirm that the use of the doughnut mode for single-beam trapping is more efficient in the axial direction.

© 1996 Optical Society of America

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References

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  1. L. Ghislain, W. Webb, “Scanning force microscope based on an optical trap,” Opt. Lett. 18, 1678–1680 (1993).
  2. H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).
  3. D. Rugar, P. Hansma, “Atomic force microscopy,” Phys. Today (23–30 October1990).
  4. G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).
  5. L. Ghislain, N. Switz, W. Webb, “Measurement of small forces using an optical trap,” Rev. Sci. Instrum. 65, 2762–2768 (1994).
  6. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–581 (1992).
  7. S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).
  8. C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).
  9. H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).
  10. J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965), pp. 330–331.

1995

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).

1994

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

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

1993

1992

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–581 (1992).

1991

S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).

1990

D. Rugar, P. Hansma, “Atomic force microscopy,” Phys. Today (23–30 October1990).

Allison, D. P.

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–581 (1992).

Bamberg, E.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Brenner, H.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965), pp. 330–331.

Butt, H.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Chen, G. Y.

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

D’Helon, C.

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

Dearden, E.

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

Engel, A.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Fendler, K.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Ghislain, L.

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

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

Goldie, K.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Hansma, P.

D. Rugar, P. Hansma, “Atomic force microscopy,” Phys. Today (23–30 October1990).

Happel, J.

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965), pp. 330–331.

He, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).

Heckenberg, N.

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

Heckenberg, N. R.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).

Inaba, H.

S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).

Ishigure, M.

S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).

Rubinsztein-Dunlop, H.

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

Rugar, D.

D. Rugar, P. Hansma, “Atomic force microscopy,” Phys. Today (23–30 October1990).

Sato, S.

S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).

Seeger, T.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Seifert, K.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Siedle, P.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Switz, N.

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

Thundat, T.

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

Warmack, R. J.

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

Webb, W.

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

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

Weisenhorn, A.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

Biophys. J.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–581 (1992).

Electron. Lett.

S. Sato, M. Ishigure, H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).

J. Microsc.

H. Butt, P. Siedle, K. Seifert, K. Fendler, T. Seeger, E. Bamberg, A. Weisenhorn, K. Goldie, A. Engel, “Scan speed limit in atomic force microscopy,” J. Microsc. 169, 75–84 (1992).

J. Mod. Opt.

C. D’Helon, E. Dearden, H. Rubinsztein-Dunlop, N. Heckenberg, “Measurement of the optical force and trapping range of a single-beam gradient optical trap for micron-sized latex spheres,” J. Mod. Opt. 41, 595–601 (1994).

H. He, N. R. Heckenberg, H. Rubinsztein-Dunlop, “Optical particle trapping with higher order doughnut beams produced using high efficiency computer generated phase holograms,” J. Mod. Opt. 42, 217–223 (1995).

Opt. Lett.

Phys. Today

D. Rugar, P. Hansma, “Atomic force microscopy,” Phys. Today (23–30 October1990).

Rev. Sci. Instrum.

G. Y. Chen, R. J. Warmack, T. Thundat, D. P. Allison, “Resonance response of scanning force microscopy cantilevers,” Rev. Sci. Instrum. 65, 2532–2537 (1994).

Rev. Sci. Instrum.

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

Other

J. Happel, H. Brenner, Low Reynolds Number Hydrodynamics (Prentice-Hall, Englewood Cliffs, N.J., 1965), pp. 330–331.

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Figures (4)

Fig. 1
Fig. 1

Generalized experimental arrangements for optical tweezers. An expanded laser beam is focused to a diffraction-limited spot by a 100× microscope objective lens. Particles are trapped in the waist of the beam. To trap using a doughnut mode, we inserted a phase hologram in the optical path in the position shown.

Fig. 2
Fig. 2

Detection of motion of trapped particle. A photodetector is positioned to intercept a portion of the backscattered light from a trapped particle. This light is focused by a system of lenses. The detector is offset from the focal plane of this system.

Fig. 3
Fig. 3

Power spectra for a 2-μm-diameter latex sphere trapped in a Gaussian beam. The bottom curve represents the power fluctuations in the laser beam with no particle trapped. The upper curves represent the theoretical (bold curve) and experimental (light curve) power spectra for a trapped particle. The spring constant for the theoretical curve is 3.81 × 106 N/m.

Fig. 4
Fig. 4

Power spectra for a 2-μm-diameter latex sphere trapped in a charge 3 doughnut beam. The bold curve represents the theoretical power spectrum and the light curve is the experimental power spectrum. The spring constant for the theoretical curve is 2.58 × 106 N/m.

Tables (1)

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Table 1 Experimental Results for Spring Constantsa

Equations (7)

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z ¨ + D m z ˙ + k z m z = B m δ ( t ) .
F = 6 π η a λ z ˙ ,
PSD = ( B / m ) 2 ( 4 π 2 f 2 - k z / m ) 2 + 4 π 2 f 2 ( D / m ) 2 ,
P av = P T ( R detector R illum ) 2 ,
Δ Z = M l Δ z = ( M t ) 2 Δ z ,
P z = d P av d z Δ z 2 P av ( M t ) 2 L - Z Δ z .
Δ X = M t Δ x .

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