Abstract

We study the axial force acting on dielectric spherical particles smaller than the trapping wavelength that are placed in the Gaussian standing wave. We derive analytical formulas for immersed particles with relative refractive indices close to unity and compare them with the numerical results obtained by generalized Lorenz–Mie theory (GLMT). We show that the axial optical force depends periodically on the particle size and that the equilibrium position of the particle alternates between the standing-wave antinodes and nodes. For certain particle sizes, gradient forces from the neighboring antinodes cancel each other and disable particle confinement. Using the GLMT we compare maximum axial trapping forces provided by the Gaussian standing-wave trap (SWT) and single-beam trap (SBT) as a function of particle size, refractive index, and beam waist size. We show that the SWT produces axial forces at least ten times stronger and permits particle confinement in a wider range of refractive indices and beam waists compared with those of the SBT.

© 2002 Optical Society of America

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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. K. Svoboda, S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
    [CrossRef] [PubMed]
  3. A. Ashkin, J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
    [CrossRef] [PubMed]
  4. S. M. Block, L. S. B. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
    [CrossRef] [PubMed]
  5. K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
    [CrossRef] [PubMed]
  6. T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
    [CrossRef] [PubMed]
  7. G. J. L. Wulte, S. B. Smith, M. Young, D. Keller, C. Bustamante, “Single-molecule studies of the effect of template tension on T7 DNA polymerase activity,” Nature 404, 103–106 (2000).
    [CrossRef]
  8. M. E. J. Friese, A. G. Truscott, H. Rubinstein-Dunlop, “Three-dimensional imaging with optical tweezers,” Appl. Opt. 38, 6597–6603 (1999).
    [CrossRef]
  9. S. Kawata, Y. Inouye, T. Sugiura, “Near-field scanning optical microscope with a laser trapped probe,” Jpn. J. Appl. Phys., Part 2 33, L1725–L1727 (1994).
    [CrossRef]
  10. K. Sasaki, H. Fujiwara, H. Masuhara, “Optical manipulation of lasing microparticle and its application to near-field microspectroscopy,” J. Vac. Sci. Technol. B 15, 2786–2790 (1997).
    [CrossRef]
  11. E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
    [CrossRef] [PubMed]
  12. A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
    [CrossRef]
  13. W. Wang, A. E. Chiou, G. J. Sonek, M. W. Berns, “Self-aligned dual-beam optical laser trap using photorefractive phase conjugation,” J. Opt. Soc. Am. B 14, 697–704 (1997).
    [CrossRef]
  14. A. Constable, J. Kim, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1867 (1993).
    [CrossRef] [PubMed]
  15. S. D. Collins, R. J. Baskin, D. G. Howitt, “Microinstrument gradient-force optical trap,” Appl. Opt. 38, 6068–6074 (1999).
    [CrossRef]
  16. K. Visscher, S. P. Gross, S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
    [CrossRef]
  17. E. Fällman, O. Axner, “Design for fully steerable dual-trap optical tweezers,” Appl. Opt. 36, 2107–2113 (1997).
    [CrossRef] [PubMed]
  18. E. R. Dufresne, D. G. Grier, “Optical tweezers arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
    [CrossRef]
  19. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. 16, 1463–1465 (1991).
    [CrossRef] [PubMed]
  20. P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of nanoparticles and microparticles using Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
    [CrossRef]
  21. Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
    [CrossRef]
  22. P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998).
    [CrossRef]
  23. M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).
  24. G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Physik 25, 377–445 (1908).
    [CrossRef]
  25. G. Gouesbet, G. Grehan, “Generalized Lorenz–Mie theories, from past to future,” Atomization Sprays 10, 277–333 (2000).
  26. J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
    [CrossRef]
  27. J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
    [CrossRef]
  28. T. Tlusty, A. Meller, R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998).
    [CrossRef]
  29. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).
  30. A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2000).
    [CrossRef]
  31. 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]
  32. G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
    [CrossRef]
  33. S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
    [CrossRef]
  34. A. Jonáš, P. Zemánek, E. L. Florin, “Single beam trapping in front of reflective surfaces,” Opt. Lett. 26, 1466–1468 (2001).
    [CrossRef]
  35. M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.

2001 (1)

2000 (3)

A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2000).
[CrossRef]

G. J. L. Wulte, S. B. Smith, M. Young, D. Keller, C. Bustamante, “Single-molecule studies of the effect of template tension on T7 DNA polymerase activity,” Nature 404, 103–106 (2000).
[CrossRef]

G. Gouesbet, G. Grehan, “Generalized Lorenz–Mie theories, from past to future,” Atomization Sprays 10, 277–333 (2000).

1999 (3)

1998 (3)

T. Tlusty, A. Meller, R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998).
[CrossRef]

P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998).
[CrossRef]

E. R. Dufresne, D. G. Grier, “Optical tweezers arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

1997 (5)

K. Sasaki, H. Fujiwara, H. Masuhara, “Optical manipulation of lasing microparticle and its application to near-field microspectroscopy,” J. Vac. Sci. Technol. B 15, 2786–2790 (1997).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

W. Wang, A. E. Chiou, G. J. Sonek, M. W. Berns, “Self-aligned dual-beam optical laser trap using photorefractive phase conjugation,” J. Opt. Soc. Am. B 14, 697–704 (1997).
[CrossRef]

E. Fällman, O. Axner, “Design for fully steerable dual-trap optical tweezers,” Appl. Opt. 36, 2107–2113 (1997).
[CrossRef] [PubMed]

1996 (3)

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

K. Visscher, S. P. Gross, S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

1994 (3)

S. Kawata, Y. Inouye, T. Sugiura, “Near-field scanning optical microscope with a laser trapped probe,” Jpn. J. Appl. Phys., Part 2 33, L1725–L1727 (1994).
[CrossRef]

T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
[CrossRef] [PubMed]

K. Svoboda, S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
[CrossRef] [PubMed]

1993 (2)

A. Constable, J. Kim, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1867–1867 (1993).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

S. M. Block, L. S. B. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

1989 (3)

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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]

1988 (1)

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

1987 (1)

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

1986 (1)

1908 (1)

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Physik 25, 377–445 (1908).
[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]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Asakura, T.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Ashkin, A.

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]

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Bar-Ziv, R.

T. Tlusty, A. Meller, R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998).
[CrossRef]

Baskin, R. J.

Berns, M. W.

Bjorkholm, J. E.

Block, S. M.

K. Visscher, S. P. Gross, S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

K. Svoboda, S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. M. Block, L. S. B. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics (Cambridge U. Press, Cambridge, UK, 1999), Chap. VIII.

Bustamante, C.

G. J. L. Wulte, S. B. Smith, M. Young, D. Keller, C. Bustamante, “Single-molecule studies of the effect of template tension on T7 DNA polymerase activity,” Nature 404, 103–106 (2000).
[CrossRef]

Chiou, A. E.

Chu, S.

T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
[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]

Collins, S. D.

Constable, A.

Dufresne, E. R.

E. R. Dufresne, D. G. Grier, “Optical tweezers arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Dziedzic, J. M.

Fällman, E.

Florin, E. L.

Florin, E.-L.

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Friese, M. E. J.

Fujiwara, H.

K. Sasaki, H. Fujiwara, H. Masuhara, “Optical manipulation of lasing microparticle and its application to near-field microspectroscopy,” J. Vac. Sci. Technol. B 15, 2786–2790 (1997).
[CrossRef]

Goldstein, L. S. B.

S. M. Block, L. S. B. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

Gouesbet, G.

G. Gouesbet, G. Grehan, “Generalized Lorenz–Mie theories, from past to future,” Atomization Sprays 10, 277–333 (2000).

G. Gouesbet, “Higher-order descriptions of Gaussian beams,” J. Opt. 27, 35–50 (1996).
[CrossRef]

Grehan, G.

G. Gouesbet, G. Grehan, “Generalized Lorenz–Mie theories, from past to future,” Atomization Sprays 10, 277–333 (2000).

Grier, D. G.

E. R. Dufresne, D. G. Grier, “Optical tweezers arrays and optical substrates created with diffractive optics,” Rev. Sci. Instrum. 69, 1974–1977 (1998).
[CrossRef]

Gross, S. P.

K. Visscher, S. P. Gross, S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

Harada, Y.

Y. Harada, T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996).
[CrossRef]

Hong, J.

A. E. Chiou, W. Wang, G. J. Sonek, J. Hong, M. W. Berns, “Interferometric optical tweezers,” Opt. Commun. 133, 7–10 (1997).
[CrossRef]

Hörber, J. K. H.

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Howitt, D. G.

Inouye, Y.

S. Kawata, Y. Inouye, T. Sugiura, “Near-field scanning optical microscope with a laser trapped probe,” Jpn. J. Appl. Phys., Part 2 33, L1725–L1727 (1994).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

Jonáš, A.

Kawata, S.

S. Kawata, Y. Inouye, T. Sugiura, “Near-field scanning optical microscope with a laser trapped probe,” Jpn. J. Appl. Phys., Part 2 33, L1725–L1727 (1994).
[CrossRef]

Keller, D.

G. J. L. Wulte, S. B. Smith, M. Young, D. Keller, C. Bustamante, “Single-molecule studies of the effect of template tension on T7 DNA polymerase activity,” Nature 404, 103–106 (2000).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969).

Kim, J.

Kitamura, N.

Koshioka, M.

Liška, M.

P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of nanoparticles and microparticles using Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998).
[CrossRef]

Masuhara, H.

K. Sasaki, H. Fujiwara, H. Masuhara, “Optical manipulation of lasing microparticle and its application to near-field microspectroscopy,” J. Vac. Sci. Technol. B 15, 2786–2790 (1997).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. 16, 1463–1465 (1991).
[CrossRef] [PubMed]

Meller, A.

T. Tlusty, A. Meller, R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998).
[CrossRef]

Mie, G.

G. Mie, “Beitrage zur Optik truber Medien, speziell kolloidaler Metallosungen,” Ann. Physik 25, 377–445 (1908).
[CrossRef]

Misawa, H.

Perkins, T. T.

T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
[CrossRef] [PubMed]

Pralle, A.

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Quake, S. R.

T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
[CrossRef] [PubMed]

Rohrbach, A.

Rubinstein-Dunlop, H.

Sasaki, K.

K. Sasaki, H. Fujiwara, H. Masuhara, “Optical manipulation of lasing microparticle and its application to near-field microspectroscopy,” J. Vac. Sci. Technol. B 15, 2786–2790 (1997).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Pattern formation and flow control of fine particles by laser-scanning micromanipulation,” Opt. Lett. 16, 1463–1465 (1991).
[CrossRef] [PubMed]

Schaub, S. A.

S. A. Schaub, J. P. Barton, D. R. Alexander, “Simplified scattering coefficient expressions for a spherical particle located on the propagation axis of a fifth-order Gaussian beam,” Appl. Phys. Lett. 55, 2709–2711 (1989).
[CrossRef]

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. P. Barton, D. R. Alexander, S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. 64, 1632–1639 (1988).
[CrossRef]

Schmidt, C. F.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

Schnapp, B. J.

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

S. M. Block, L. S. B. Goldstein, B. J. Schnapp, “Bead movement by single kinesin molecules studied with optical tweezers,” Nature 348, 348–352 (1990).
[CrossRef] [PubMed]

Smith, D. E.

T. T. Perkins, S. R. Quake, D. E. Smith, S. Chu, “Relaxation of a single DNA molecule observed by optical microscopy,” Science 264, 822–825 (1994).
[CrossRef] [PubMed]

Smith, S. B.

G. J. L. Wulte, S. B. Smith, M. Young, D. Keller, C. Bustamante, “Single-molecule studies of the effect of template tension on T7 DNA polymerase activity,” Nature 404, 103–106 (2000).
[CrossRef]

Sonek, G. J.

Šrámek, L.

P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of nanoparticles and microparticles using Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

P. Zemánek, A. Jonáš, L. Šrámek, M. Liška, “Optical trapping of Rayleigh particles using a Gaussian standing wave,” Opt. Commun. 151, 273–285 (1998).
[CrossRef]

Stelzer, E. H. K.

A. Rohrbach, E. H. K. Stelzer, “Optical trapping of dielectric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2000).
[CrossRef]

E.-L. Florin, A. Pralle, J. K. H. Hörber, E. H. K. Stelzer, “Photonic force microscope based on optical tweezers and two-photon excitation for biological applications,” J. Struct. Biol. 119, 202–211 (1997).
[CrossRef] [PubMed]

Sugiura, T.

S. Kawata, Y. Inouye, T. Sugiura, “Near-field scanning optical microscope with a laser trapped probe,” Jpn. J. Appl. Phys., Part 2 33, L1725–L1727 (1994).
[CrossRef]

Svoboda, K.

K. Svoboda, S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
[CrossRef] [PubMed]

K. Svoboda, C. F. Schmidt, B. J. Schnapp, S. M. Block, “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993).
[CrossRef] [PubMed]

Tlusty, T.

T. Tlusty, A. Meller, R. Bar-Ziv, “Optical gradient forces of strongly localized fields,” Phys. Rev. Lett. 81, 1738–1741 (1998).
[CrossRef]

Truscott, A. G.

Visscher, K.

K. Visscher, S. P. Gross, S. M. Block, “Construction of multiple-beam optical traps with nanometer-resolution position sensing,” IEEE J. Sel. Top. Quantum Electron. 2, 1066–1076 (1996).
[CrossRef]

Wang, W.

Wolf, E.

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Figures (7)

Fig. 1
Fig. 1

Orientation of axes in the standing-wave apparatus. The z axis follows the direction of the reflected wave. Positive z0 means that the beam waist of radius w0 is located in the reflected wave.

Fig. 2
Fig. 2

Profile of the size-dependent term Ga in the analytical form of the axial force acting on a dielectric sphere in the GSW.

Fig. 3
Fig. 3

Behavior of two polystyrene spheres of slightly different radii (a=0.3λ, 0.35λ) placed symmetrically with respect to the standing-wave node. The following parameters were used for the force calculation: w0=λ, ρ=1, ψ=3π/2, z0=0 µm, m=1.95, P=1 W, λvac=1064 nm.

Fig. 4
Fig. 4

Maximum relative error of the ESA methods (in percent) as a function of the sphere radius and relative refractive index for four beam waist sizes (w0=0.75, 1, 1.25, 2.5λ) and the following parameters: ρ=1, ψ=3π/2, λvac=1064 nm, z0=0 µm.

Fig. 5
Fig. 5

Maximum relative error of the ESA methods (in percent) if the α=m2-1 term is replaced by 3(m2-1)/(m2+2) as a function of the sphere radius and relative refractive index for four beam waist sizes (w0=0.75, 1, 1.25, 2.5λ) and the following parameters: ρ=1, ψ=3π/2, λvac=1064 nm, z0=0 µm.

Fig. 6
Fig. 6

Contour plot of maximum SWT axial trapping force (in piconewtons) with respect to the z axis as a function of the relative refractive index, particle size, and beam waist radius calculated by using the GLMT. Dotted–dashed and dashed lines represent the loci of maximum and minimum values of the maximum trapping force with respect to the particle size and relative refractive index. The shaded areas represent the nontrapping regions. The following parameters were used: P=1 W, ρ==1, ψ=3π/2, λvac=1064 nm, z0=0 µm.

Fig. 7
Fig. 7

Contour plot of maximum SBT axial trapping force (in piconewtons) with respect to the z axis as a function of relative refractive index, particle size, and beam waist radius calculated by using the GLMT. The dotted–dashed curves represent the loci of maximum and minimum vales of the maximum trapping force with respect to particle size and relative refractive index. The dotted curves denote the borders of the trapping regions, and the shaded areas represent the nontrapping regions. The inset plots in the bottom two plots magnify the regions of interest with the same horizontal scale. The following parameters were used: P=1 W, λvac=1064 nm.

Equations (52)

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ΔW=12V(ED-E0D0+HB-H0B0)dV,
ΔW(r, t)T=-122αV1E(r, t)E0(r, t)TdV,
ΔW(r, t)T=-122αV1|E0(r, t)|2TdV.
I0(r)S(r, t)TE0(r, t)×H0(r, t)T,
z0n20c|E0(r, t)|2T=z0n20c|E0(r)|2/2=z0I0(r).
ΔW(r)ΔW(r, t)T=-α2n2cV1I0(r)dV.
F(r)-ΔW(r, t)T=α2n2cV1I0(r)dV=α2n2cS1nI0(r)dS,
Ei(rB, zB)=E00w0wiexp-rB2wi2expik(zB+z0)-i2krB2Ri-i arctan (zB+z0zR),
Er(rB, zB)=E00ρw0wrexp-rB2wr2exp-ik(zB-z0)-i2krB2Rr+i arctanzB-z0zR-iψ,
wi/r(zB)=w01+(zB±z0)2zR21/2,
Ri/r(zB)=(zB±z0)1+zR2(zB±z0)2,
IGSW(rB, zB)=n20c2|Ei(rB, zB)+Er(rB, zB)|2
=I00w02wi2exp[-(2rB2/wi2)]+2ρI00w02wiwrexp[-(rB2/wi2)]×exp[-(rB2/wr2)]cos ϕGSW+ρ2I00w02wr2exp[-(2rB2/wr2)],
ϕGSW(rB, zB)=2kzB-krB221Ri-1Rr-arctanzB+z0zR-arctanzB-z0zR+ψ.
ΔW(rs, zs)=-α2n2c2π0a0πIGSW(rB, zB)r2 sin χdχdr,
Fz(rs, zs)=α2n2c2π0πIGSW(rB, zB)a2 sin χ cos χdχ.
ΔW(rs, zs)=-23n2cαπa3IGSW(rs, zs),
F(rs, zs)=23n2cαπa3IGSW(rs, zs).
α=m2-13[(m2-1)/(m2+2)],
ΔW(rs, zs)=-αn2cP1wi2exp[-(2rs2/wi2)]+ρ21wr2exp[-(2rs2/wr2)]43a3+ρk3wiwrexp[-rs2(1/wi2+1/wr2)]×(sin 2ka-2ka cos 2ka)cos ϕs,
Fz(rs, zs)=-2n2αcPk2wiwrρ exp[-rs2(1/wi2+1/wr2)]×(sin 2ka-2ka cos 2ka)sin ϕs,
ΔW(0, zs)
=-αn2cPa(1+ρ2)-wi2daw2awi+ρ2wr2daw2awr+ρwiwr2W2k(1+ZW2)sin(2ka)(cos ϕs-ZW sin ϕs)+Re2W3C3/2exp(iϕs)[exp(i2ka)daw(X+Y)+exp(-i2ka)daw(-X+Y)],
Fz(0, zs)
=4αn2cPρwiwr-W21+ZW2sin (2ka)×(sin ϕs+ZW cos ϕs)+ReikW3C3/2exp(iϕs)[exp(i2ka)daw(X+Y)+exp(-i2ka)daw(-X+Y)],
ΔFz=max(|Fapr(azsa+λ)-FGLMT(azsa+λ)|)Fmax,
ΔW=12V(ED0-DE0)dV+12V(E+E0)(D-D0)dV.
12V(E+E0)(D-D0)dV=12V-ϕ0-A0t-ϕ-At(D-D0)dV,=12V-(ϕ0+ϕ)-(A0+A)t(D-D0)dV,=-12VϕT(D-D0)dV-12VATt(D-D0)dV,
-12V(AT0 sin ωt)t(D-D0)0 sin ωt dV=-12VωAT0(D-D0)0 cos ωt sin ωt dV.
-12VATt(D-D0)dVT=0.
-12VϕT(D-D0)dV=-12V[ϕT(D-D0)]dV+12VϕT(D-D0)dV.
-12V[ϕT(D-D0)]dV=-12S[ϕT(D-D0)]ndS.
ΔWET=-12Vi(2-1)EE0TdV.
2rB2wi/r2=2rs2wi/r2+2rsr sin χ cos ϕwi/r2+r2 sin2 χwi/r22rs2wi/r2,
wi/r2=w021+zs±z0+r cos χzR2w021+zs±z0zR2,
k21Ri/r=k2zs±z0+r cos χzR2[1+(zs±z0+r cos ξ)2/zR2]zs±z0zRwi/r2,
krB221Ri-1Rr-rs2zs+z0zRwi2+zs-z0zRwr2.
Fz(rs, zn)=α2n2cI004πρw02wiwra2 exp(-rs2/wi2-rs2/wr2)-11 cos(2kaξ+ϕs)ξdξdr,
=αn2cI00πρw02k2wiwrexp(-rs2/wi2-rs2/wr2)(2ak cos 2ak-sin 2ak)sin ϕs.
2rB2wt/r2=2r2 sin2 χwi/r2,
krB221Ri-1Rr-r2 sin2 χzs+z0zRwi2+zs-z0zRwr2.
Fz(0, zs)=αn2cI00ρ2πa2w02wiwr-11 exp[-a2(1-ξ2)/W2]×cosϕs+2akξ+a2R02(1-ξ2)ξdξ,
=αn2cI00ρ2πa2w02wiwrReexp(iϕs)×-11exp{[-a2C(1-ξ2)/W2]+2iakξ}ξdξ,
=αn2cI00ρ2πw02wiwr-W21+ZW2sin (2ak)×(sin ϕs+ZW cos ϕs)+RekW3π2C3/2exp(iϕs-X2-Y2)×[erf(iX+iY)+erf(-iX+iY)],
W=wi(zs)wr(zs)/[wi2(zs)+wr2(zs)]1/2,
X=-kW/C,
Y=aC/W,
C=1-iZW,
ZW=W2/R02,
1/R02=(zs+z0)/(zRwi2)+(zs+z0)/(zRwr2).
erf(ix)=2iπexp(x2)daw(x),
daw(x)=0xexp(t2-x2)dt,

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