Abstract

Optical tweezers, which utilize radiation pressure to control and manipulate microscopic particles, are used for a large number of applications in biology and colloid science. In most applications a single optical tweezers is used to control one single particle. However, two or more particles can be trapped simultaneously. Although this characteristic has been used in applications, no theoretical analysis of the trapping force or the status of the trapped particles is available to our knowledge. We present our calculation, using a ray optics model, of the axial trapping forces on two rigid particles trapped in optical tweezers. The spherical aberration that results from a mismatch of the refractive indices of oil and water is also considered. The results show that the forces exerted by the optical tweezers on the two particles will cause the two particles to touch each other, and the two particles can be stably trapped at a joint equilibrium point. We also discuss the stability of axial trapping. The calculation will be useful in applications of optical tweezers to trap multiple particles.

© 2005 Optical Society of America

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References

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  1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  2. A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
    [CrossRef] [PubMed]
  3. A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
    [CrossRef] [PubMed]
  4. M. D. Wang, “Manipulation of single molecules in biology,” Curr. Opin. Biotechnol. 10, 81–86 (1999).
    [CrossRef] [PubMed]
  5. J. C. Crocker, “Measurement of the hydrodynamic corrections to the Brownian motion of two colloidal spheres,” J. Chem. Phys. 106, 2837–2840 (1997).
    [CrossRef]
  6. J. C. Crocker, D. G. Grier, “Microscopic measurement of the pair interaction potential of charge-stabilized colloid,” Phys. Rev. Lett. 73, 352–355 (1994).
    [CrossRef] [PubMed]
  7. D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
    [CrossRef]
  8. R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
    [CrossRef]
  9. A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
  10. G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  11. R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
    [CrossRef]
  12. J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  13. A. Rohrbach, E. H. K. Stelzer, “Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations,” Appl. Opt. 41, 2494–2507 (2002).
    [CrossRef] [PubMed]
  14. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
    [CrossRef]
  15. X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
    [CrossRef]
  16. E. Fallman, O. Axner, “Influence of a glass–water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003).
    [CrossRef]
  17. V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
    [CrossRef] [PubMed]
  18. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef] [PubMed]
  19. Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
    [CrossRef]

2003

E. Fallman, O. Axner, “Influence of a glass–water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42, 3915–3926 (2003).
[CrossRef]

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

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

2002

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

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

2001

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

1999

M. D. Wang, “Manipulation of single molecules in biology,” Curr. Opin. Biotechnol. 10, 81–86 (1999).
[CrossRef] [PubMed]

1998

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

1997

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

J. C. Crocker, “Measurement of the hydrodynamic corrections to the Brownian motion of two colloidal spheres,” J. Chem. Phys. 106, 2837–2840 (1997).
[CrossRef]

1994

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

1992

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
[CrossRef]

1989

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

1987

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

1986

1979

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

Alexander, D. R.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

Axner, O.

Barton, J. P.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Bjorkholm, J. E.

Brevik, I.

Cheng, B. Y.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Chu, S.

Crocker, J. C.

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

J. C. Crocker, “Measurement of the hydrodynamic corrections to the Brownian motion of two colloidal spheres,” J. Chem. Phys. 106, 2837–2840 (1997).
[CrossRef]

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

Dai, G. L.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Dholakia, K.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Dziedzic, J. M.

Fallman, E.

Funatsu, T.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Garces-Chavez, V.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Grier, D. G.

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

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

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

Guo, H. L.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Gussgard, R.

Heckenberg, N. R.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

Higuchi, H.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Ishijima, A.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Kojima, H.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Li, Y. M.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Li, Z. L.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Lindmo, T.

Liu, Q. S.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Lou, L. R.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

McGloin, D.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Nieminen, T. A.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

Owen, R. J.

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

Rohrbach, A.

Roosen, G.

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

Rubinsztein-Dunlop, H.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

Schaub, S. A.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Sibbett, W.

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

Stelzer, E. H. K.

Sun, Z. W.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Tanaka, H.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Tokunaga, M.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Verma, R.

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

Wang, M. D.

M. D. Wang, “Manipulation of single molecules in biology,” Curr. Opin. Biotechnol. 10, 81–86 (1999).
[CrossRef] [PubMed]

Xu, S. H.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Yanagida, T.

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Yao, X. C.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Yodh, A. G.

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

Zhang, D. Z.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Zhu, R. Z.

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

Appl. Opt.

Biophys. J.

A. Ashkin, “Forces of a single-beam gradient trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).

Can. J. Phys.

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

Cell

A. Ishijima, H. Kojima, T. Funatsu, M. Tokunaga, H. Higuchi, H. Tanaka, T. Yanagida, “Simultaneous observation of individual ATPase and mechanical events by a single myosin molecule during interaction with actin,” Cell 92, 161–171 (1998).
[CrossRef] [PubMed]

Chin. Phys. Lett.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, D. Z. Zhang, “Effects of spherical aberration on optical trapping forces for Rayleigh particles,” Chin. Phys. Lett. 18, 432–434 (2001).
[CrossRef]

Curr. Opin. Biotechnol.

M. D. Wang, “Manipulation of single molecules in biology,” Curr. Opin. Biotechnol. 10, 81–86 (1999).
[CrossRef] [PubMed]

Curr. Opin. Colloid Interface Sci.

D. G. Grier, “Optical tweezers in colloid and interface science,” Curr. Opin. Colloid Interface Sci. 2, 264–270 (1997).
[CrossRef]

J. Appl. Phys.

J. P. Barton, D. R. Alexander, S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. Chem. Phys.

J. C. Crocker, “Measurement of the hydrodynamic corrections to the Brownian motion of two colloidal spheres,” J. Chem. Phys. 106, 2837–2840 (1997).
[CrossRef]

Z. W. Sun, S. H. Xu, G. L. Dai, Y. M. Li, L. R. Lou, Q. S. Liu, R. Z. Zhu, “A microscopic approach to studying colloidal stability,” J. Chem. Phys. 119, 2399–2405 (2003).
[CrossRef]

J. Opt. Soc. Am. B

J. Quant. Spectrosc. Radiat. Transfer

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation and optical measurement of laser trapping forces on non-spherical particles,” J. Quant. Spectrosc. Radiat. Transfer 70, 627–637 (2001).
[CrossRef]

Nature

V. Garces-Chavez, D. McGloin, H. Melville, W. Sibbett, K. Dholakia, “Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam,” Nature 419, 145–147 (2002).
[CrossRef] [PubMed]

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

Opt. Lett.

Phys. Rev. E.

R. J. Owen, J. C. Crocker, R. Verma, A. G. Yodh, “Measurement of long-range steric repulsions between microspheres due to an adsorbed polymer,” Phys. Rev. E. 64, 11401 (2001).
[CrossRef]

Phys. Rev. Lett.

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

Science

A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

RO model for optical trapping (after Ashkin9). Scattering force Fs points in the direction of the incident ray; gradient force Fg is perpendicular to the direction of the incident ray. θ is the angle of incidence; r is the angle of refraction. The intensities of the reflected and refracted rays are determined by Fresnel reflection R and transmission T coefficients, respectively, and by power p of the incident ray. The parameters of the reflected and transmitted rays out of the bead can also be determined from this figure.

Fig. 2
Fig. 2

Schematic illustration of light in an optical tweezers system including a glass–water interface (after Fallman and Axner16). The parameters marked in the figure are needed in the calculation and are discussed in the text.

Fig. 3
Fig. 3

Trapping force exerted on the second particle if the first particle is assumed to be trapped at its own equilibrium point and is not in contact with the second particle. The leftmost limits of the curves are the equilibrium points of the first particle plus the diameter of the particle. (a) zcg = −3rb; (b) zcg = −4rb.

Fig. 4
Fig. 4

Trapping force exerted on the second particle. The leftmost limits of the curves are the equilibrium points of the first particle plus the diameter of the particle. (a), (c) zcg = −3rb; (b), (d) zcg = −4rb. (a), (b) The first particle stays at the position corresponding to joint equilibrium. (c), (d) The first particle is always in contact with the second particle.

Fig. 5
Fig. 5

Curves of the trapping potential of the second particle; the power of the laser beam is assumed to be 10 mW. The potentials on the positions where the two particles are at equilibrium points are taken to be 0. (a), (c) zcg = −3rb; (b), (d) zcg = −4rb. (a), (b) The first particle stays at the position corresponding to joint equilibrium. (c), (d) The first particle is always in contact with the second particle.

Equations (7)

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

Q s = 1 + R cos ( 2 θ ) T 2 [ cos ( 2 θ 2 r ) + R cos ( 2 θ ) ] 1 + R 2 + 2 R cos ( 2 r ) ,
Q g = R sin ( 2 θ ) T 2 [ cos ( 2 θ 2 r ) + R sin ( 2 θ ) ] 1 + R 2 + 2 R cos ( 2 r ) ,
sin ( θ ) = z b Δ z r b sin ( ϕ ) .
Δ z = z c g { 1 n w n g [ 1 η 2 ( NA / n w ) 2 1 η 2 ( NA / n g ) 2 ] 1 / 2 } ,
η = ρ / ρ p .
sin ( ϕ ) = NA n w η .
I ( ρ ) = { I 0 exp ( 2 ρ 2 / w 0 2 ) ρ ρ p 0 ρ > ρ p ,

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