Abstract

We present a new method to calculate trapping forces of dielectric particles with diameters Dλ in arbitrary electromagnetic, time-invariant fields. The two components of the optical force, the gradient force and the scattering force, are determined separately. Both the arbitrary incident field and the scatterer are represented by plane-wave spectra. The scattering force is determined by means of the momentum transfer in either single- or double-scattering processes. Therefore the second-order Born series is evaluated and solved in the frequency domain by Ewald constructions. Numerical results of our two-force-component approach and an established calculation method are compared and show satisfying agreement. Our procedure is applied to investigate axial trapping by focused waves experiencing effects of aperture illumination and refractive-index mismatch.

© 2001 Optical Society of America

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1999 (2)

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

P. Zemánek, A. Jonás, L. Srámek, M. Liska, “Opticaltrapping of nanoparticles and microparticles by a Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

1998 (5)

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

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

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

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

A. Rohrbach, W. Singer, “Scattering of a scalar field at dielectric surfaces by Born series expansion,” J. Opt. Soc. Am. A 15, 2651–2659 (1998).
[CrossRef]

1997 (4)

T. Lemaire, “Coupled-multipole formulation for the treatment of electromagnetic scattering by a small dielectric particle of arbitrary shape,” J. Opt. Soc. Am. A 14, 470–474 (1997).
[CrossRef]

M. Gu, P. C. Ke, X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666–3668 (1997).
[CrossRef]

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

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

1996 (2)

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

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

1995 (3)

1994 (3)

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

B. T. Draine, P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1498 (1994).
[CrossRef]

1993 (4)

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (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]

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

1992 (3)

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]

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

K. Visscher, G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik 89, 174–180 (1992).

1991 (2)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

1990 (2)

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

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

1989 (5)

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (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, “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]

M. Mansuripur, “Certain computational aspects of vector diffraction problems,” J. Opt. Soc. Am. A 6, 786–805 (1989).
[CrossRef]

1987 (1)

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

1986 (2)

1985 (1)

1983 (1)

1979 (2)

S. Colak, C. Yeh, L. W. Casperson, “Scattering of focused beams by tenuous particles,” Appl. Opt. 18, 294–302 (1979).
[CrossRef] [PubMed]

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

1976 (1)

1973 (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

1972 (1)

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1964 (1)

1919 (1)

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebige Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beitraege zur Optik trueber Medien speziell Kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Acquista, C.

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]

Allersma, M. W.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

Andrews, J. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[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.

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

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

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]

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Barton, J. P.

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, “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]

Berg, H. C.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Berns, M. W.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Bjorkholm, J. E.

Blair, D. F.

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Block, S. M.

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

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Bohren, C.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paperback, New York, 1998), p. 137.

Booker, G. R.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999).

Brakenhoff, G. J.

K. Visscher, G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik 89, 174–180 (1992).

Brenner, K.-H.

Brevik, I.

Casperson, L. W.

Cheng, S.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Chu, S.

Colak, S.

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebige Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

deCastro, M. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

Draine, B. T.

Dziedzic, J. M.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

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]

Euteneuer, U.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

Felgner, H.

Flatau, P. J.

Florin, E.-L.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

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

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

Futterman, G.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Gan, X. S.

M. Gu, P. C. Ke, X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666–3668 (1997).
[CrossRef]

Ghislain, L. P.

Gittes, F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

Goldstein, L. S.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, Calif., 1968), pp. 48–56.

Gordon, J. P.

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

Gouesbet, G.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Gréhan, G.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Greulich, K. O.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Gu, M.

M. Gu, P. C. Ke, X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666–3668 (1997).
[CrossRef]

Gussgard, R.

Harada, Y.

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

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Hörber, J. K. H.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

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

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

Huffman, D. R.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paperback, New York, 1998), p. 137.

Hutter, K. J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 236.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 239.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 151.

Jonás, A.

P. Zemánek, A. Jonás, L. Srámek, M. Liska, “Opticaltrapping of nanoparticles and microparticles by a Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

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

Ke, P. C.

M. Gu, P. C. Ke, X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666–3668 (1997).
[CrossRef]

Kerker, M.

M. Kerker, The Scattering of Light, 1st ed. (Academic, New York, 1969).

M. Kerker, “Rayleigh–Debye scattering,” in The Scattering of Light, E. M. Loebl, ed., 1st ed. (Academic, New York, 1969), p. 414.

Kim, J. S.

Kuo, S. C.

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Laczik, Z.

Lalor, E.

Larkin, K. G.

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

Lee, S. S.

Lemaire, T.

Lindmo, T.

Liska, M.

P. Zemánek, A. Jonás, L. Srámek, M. Liska, “Opticaltrapping of nanoparticles and microparticles by a Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

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

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Mansuripur, M.

McCutchen, C. W.

Mie, G.

G. Mie, “Beitraege zur Optik trueber Medien speziell Kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Monajembashi, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Müller, O.

Mulser, P.

Numajiri, Y.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Pralle, A.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

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

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

Profeta, G. A.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Prummer, M.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

Ren, F.

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

Rohrbach, A.

Rosin, A.

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

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]

Schliwa, M.

H. Felgner, O. Müller, M. Schliwa, “Calibration of light forces in optical tweezers,” Appl. Opt. 34, 977–982 (1995).
[CrossRef] [PubMed]

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

Schmidt, C. F.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[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]

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

Schutze, K.

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

Seeger, S.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Sheetz, M. P.

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Sheppard, C. J. R.

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

Shifrin, K. S.

K. S. Shifrin, Scattering of Light in a Turbid Medium, 1st ed. (Nauka, Moscow, 1951) N. T. t. T. F.-. (1968).

Singer, W.

Sonek, G. J.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

Srámek, L.

P. Zemánek, A. Jonás, L. Srámek, M. Liska, “Opticaltrapping of nanoparticles and microparticles by a Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

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

Stelzer, E. H. K.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

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

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

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

Steubing, R. W.

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

Stewart, R. J.

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

Svoboda, K.

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]

Török, P.

Tromberg, B. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Varga, P.

Visscher, K.

K. Visscher, G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik 89, 174–180 (1992).

Walter, R. J.

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Webb, W. W.

Weyl, H.

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

Wohland, T.

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

Wolf, E.

E. Lalor, E. Wolf, “Exact solution of the equation of molecular optics for refraction and reflection of an electromagnetic wave on a semi-infinite dielectric,” J. Opt. Soc. Am. 62, 1165–1174 (1972).
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M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Wolfrum, J.

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

Wright, W. H.

W. H. Wright, G. J. Sonek, M. W. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
[CrossRef] [PubMed]

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Yeh, C.

Zemánek, P.

P. Zemánek, A. Jonás, L. Srámek, M. Liska, “Opticaltrapping of nanoparticles and microparticles by a Gaussian standing wave,” Opt. Lett. 24, 1448–1450 (1999).
[CrossRef]

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

Am. J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am. J. Phys. 47, 974–980 (1979).
[CrossRef]

Ann. Phys. (3)

P. Debye, “Der Lichtdruck auf Kugeln von beliebige Material,” Ann. Phys. 30, 57–136 (1909).
[CrossRef]

G. Mie, “Beitraege zur Optik trueber Medien speziell Kolloidaler Metalloesungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

H. Weyl, “Ausbreitung elektromagnetischer Wellen ueber einem ebenen Leiter,” Ann. Phys. 60, 481–500 (1919).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. A (1)

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, P71–P73 (1998).
[CrossRef]

Appl. Phys. A: Solids Surf. (1)

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

Appl. Phys. Lett. (1)

W. H. Wright, G. J. Sonek, M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. 63, 715–717 (1993).
[CrossRef]

Biophys. J. (2)

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

M. W. Allersma, F. Gittes, M. J. deCastro, R. J. Stewart, C. F. Schmidt, “Two-dimensional tracking of ncd motility by back focal plane interferometry,” Biophys. J. 74, 1074–1085 (1998).
[CrossRef] [PubMed]

Cytometry (2)

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[CrossRef] [PubMed]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

J. Appl. Phys. (2)

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, “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. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (7)

J. Opt. Soc. Am. B (2)

J. Struct. Biol. (1)

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

Microsc. Res. Tech. (1)

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional position tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [PubMed]

Nature (1)

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]

Nature (London) (3)

S. M. Block, D. F. Blair, H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

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

A. Ashkin, K. Schutze, J. M. Dziedzic, U. Euteneuer, M. Schliwa, “Force generation of organelle transport measured in vivo by an infrared laser trap,” Nature (London) 348, 346–348 (1990).
[CrossRef]

Opt. Commun. (3)

F. Ren, G. Gréhan, G. Gouesbet, “Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorentz–Mie theory, and associated resonance effects,” Opt. Commun. 108, 343–354 (1994).
[CrossRef]

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

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

Opt. Lett. (3)

Optik (3)

T. Wohland, A. Rosin, E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik 102, 181–190 (1996).

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

K. Visscher, G. J. Brakenhoff, “Theoretical study of optically induced forces on spherical particles in a single beam trap I: Rayleigh scatterers,” Optik 89, 174–180 (1992).

Phys. Rev. (1)

J. P. Gordon, “Radiation forces and momenta in dielectric media,” Phys. Rev. 8, 14–21 (1973).
[CrossRef]

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A. (1)

M. W. Berns, W. H. Wright, B. J. Tromberg, G. A. Profeta, J. J. Andrews, R. J. Walter, “Use of a laser-induced optical force trap to study chromosome movement on the mitotic spindle,” Proc. Natl. Acad. Sci. U.S.A. 86, 7914–7918 (1989).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Gu, P. C. Ke, X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666–3668 (1997).
[CrossRef]

Science (2)

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

S. C. Kuo, M. P. Sheetz, “Force of single kinesin molecules measured with optical tweezers,” Science 260, 232–234 (1993).
[CrossRef] [PubMed]

Other (12)

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 236.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 239.

J. D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975), p. 151.

C. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Science Paperback, New York, 1998), p. 137.

A. Rohrbach, www.embl-heidelberg.de/∼rohrbach .

M. Kerker, The Scattering of Light, 1st ed. (Academic, New York, 1969).

K. S. Shifrin, Scattering of Light in a Turbid Medium, 1st ed. (Nauka, Moscow, 1951) N. T. t. T. F.-. (1968).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995).

Mie scattering, www.omlc.ogi.edu/calc/mie_calc.html .

J. W. Goodman, Introduction to Fourier Optics, 1st ed. (McGraw-Hill, San Francisco, Calif., 1968), pp. 48–56.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, New York, 1999).

M. Kerker, “Rayleigh–Debye scattering,” in The Scattering of Light, E. M. Loebl, ed., 1st ed. (Academic, New York, 1969), p. 414.

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Figures (8)

Fig. 1
Fig. 1

Coordinate system of propagation and scattering: a dielectric spherical object with permittivity εs and radius a is located in a medium with permittivity εm. The object is shifted by the vector b=(bx, by, bz) from the origin (0, 0, 0). The electric field’s x component Exm (left) and z component Ezm (right) of a highly focused wave, traveling along the z direction are shown in the background. Bright areas indicate a larger field amplitude.

Fig. 2
Fig. 2

First (second-) order momentum transfer from a dielectric sphere onto the incoming field (onto the first-order scattered field). The spectrum of plane waves E˜(-ki) of the incoming field is indicated by the bold circle segment with half-angle sin(α)=NA/nm. The momentum spectrum of the sphere s˜(kx, ky, kz) is shown as the background, where dark indicates a larger transfer probability. The first-order momentum transfer for a single plane wave obeys g=ks(1)-ki, with transfer strength E˜(-ki)s˜(ks(1)-ki); the second-order momentum transfer is given by ks(2)-ks(1), with transfer strength E˜(-ki)s˜(ks(1)-ki)s˜(ks(2)-ks(1)). Owing to their constant length, all wave vectors ks end on Ewald spheres, all having the same radius (coherent scattering). This is indicated by a solid circle for first-order scattering and by a dashed circle for second-order scattering.

Fig. 3
Fig. 3

Comparison of scattering efficiencies Qsca=Csca/(a2π) for different sphere radii a and refractive indices ns computed with Mie theory and with the second-order Born approximation. The scattering efficiencies correspond to the scattering of a plane wave in water (λ0=1.064 µm, nm=1.33) by nonabsorbing dielectric spheres.

Fig. 4
Fig. 4

Comparison of the intensity distribution of a highly focused, x-polarized beam calculated with fifth-order corrected Gaussian beam optics and with our exact method. The immersion medium is water (nm=1.33), the wavelength is λ0=1.064 µm. The line scans in x,  y, and z correspond to an ideal focus of an NA=1.2 lens. The circular lens aperture with radius R was illuminated by a Gaussian beam with waist w=0.8R.

Fig. 5
Fig. 5

Comparison of results derived by different calculation methods for axial forces as a function of axial position. A homogeneous dielectric particle (a=100 nm, ns=1.57) is moved along the z axis through the focus of an x-polarized Gaussian beam with w0x=0.59 µm. The wavelength is λ0=1.064 µm in vacuum, and the refractive index in the medium is nm=1.33. The solid curve shows the axial normalized trapping force Q (i.e., trapping efficiency) determined by Maxwells stress tensor and Mie theory; the dashed curve shows the result of our computation with the two-component procedure: It is the sum of the scattering (dotted–dashed curve) and the gradient (dotted curve) forces. Positive values of Q denote forces pushing the sphere in the propagation direction of the light; a negative Q denotes a force that pulls in the opposite direction.

Fig. 6
Fig. 6

Maximum backward trapping efficiency and trapping force of a sphere as a function of the radius. Spheres with indices n=1.46 (fused silica) and n=1.57 (latex) are trapped by a strongly focused x-polarized beam. The Gaussian beam waist in x is w0x=0.52 µm at a wavelength of λn=1.064 µm/1.33=0.800 µm in water. The solid curves represent the result of a Mie theory computation; the dashed curves are results obtained with the two-component procedure. Corresponding forces are in the piconewton range at a laser power of P=100 mW.

Fig. 7
Fig. 7

Influence of refractive-index mismatch and of an aperture overillumination on the maximum backward trapping efficiency as a function of sphere radius. The difference is given relative to the case shown in Fig. 6 (a Gaussian illumination with a beam waist w0=0.8R, no refractive-index mismatch). The curves are plotted as follows: For no SA, but a Gaussian illumination with beam waist w0=2.0R (solid curves) (R=aperture radius), for SA at d=10 µm, Gaussian illumination with beam waist w0=0.8R (dashed curves), and for beam waist w0=0.8R, no SA, first-order Born approximation for the scattering force (dotted curve). Left, spheres with index ns=1.46; right, spheres with index ns=1.57.

Fig. 8
Fig. 8

Two-dimensional axial trapping potential for a spherical particle (a=100 nm, ns=1.57) in a focus from an ideal lens with NA=1.2. The asymmetry in the z direction can be seen clearly. To escape from the trap, a trapped particle must have an energy equal to the difference between its thermal energy and the energy minimum in the z=+3µm plane. The strength of the potential is scaled in atto-Joule (aJ=10-18 J) for an incident-light power of 100 mW.

Equations (67)

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2E(r)+n(r)2k02E(r)=0.
n(r)2=nm2-Δn(r)2.
E˜(kx, ky, kz)=E(x, y, z)×exp[i(kxx+kyy+kzz)]dxdydz.
(|k|2-kn2)E˜(k)=0|k|2=kx2+ky2+kz2=kn2.
k=(kx2+ky2)1/2=kn sin(θ),
kz=(kn2-kx2-ky2)1/2=kn cos(θ),
E˜(kx, ky, kz)=E˜+(kx, ky)δ[kz-(kn2-kx2-ky2)1/2]+E˜-(kx, ky)δ[kz+(kn2-kx2-ky2)1/2].
E˜+(kx, ky)=0E˜(kx, ky, kz)dkz,
E˜-(kx, ky)=-0E˜(kx, ky, kz)dkz.
Ef(x, y, z)=1(2π)3A[kx,ky, (kn2-kx2-ky2)1/2]×exp(-ikr)dkxdkydkz.
A(k)=E0(k)T(k)P(k)B(k)=A+(kx, ky)δ[kz-(kn2-kx2-ky2)1/2].
B(k)=(kn/kz)1/2=cos(θ)-1/2.
Px(k)=(kx2kz/kn+ky2)/k2,
Py(k)=-kxky[1-(kz/kn)]/k2,
Pz(k)=kx/kn.
E˜(kx, ky, f+Δz)=E˜(kx, ky, f)exp[-iΔzkz(kx, ky)].
f(r, t)=[P(r, t)]Em(r, t)+P(r, t)t×Bm(r, t),
P(r, t)=εmαEm(r, t)
f(r, t)=αεm(|Em(r, t)|2/2)+αεmt[Em(r, t)×Bm(r, t)].
S(r, t)=nmε0c2[Em(r, t)×Bm(r, t)]
I(r)=|S(r, t)|T=nmε0c2/2|Em(r)×Bm(r)|=nmε0c/2|Em(r)|2.
m(r, t)=S(r, t)/c2=εm/nm[Em(r, t)×Bm(r, t)].
Δm(r)=[m(r, tafter)T-m(r, tbefore)T].
f(r)=αnm2cIm(r)+αnmΔm(r)Δt=fgrad(r)+fscat(r).
Eint=[(3εm)/(εs+2εm)]Em.
p=PdV=4πa3εm[(εs-εm)/(εs+2εm)]Em.
α=3(εs-εm)/(εs+2εm)=3(m2-1)/(m2+2).
g=ks-ki,ks=ks(1)+ks(2)+.
(2+nm2k02)E(r)=Δn2s(r)k02E(r)=(nm2-ns2)s(r)k02Eint(r)=αs(r)nm2k02E(r).
E(r)=Em(r)+αkn2s(r)E(r)G(r, r)d3r=Em(r)+Es(r).
E(r)=p=0αpEs(p)(r)=Es(0)(r)+[αEs(1)(r)+α2Es(2)(r)+].
Es(r)=αkn2s(r)Es(0)(r)G(r, r)d3r+α2kn2s(r)Es(1)(r)G(r, r)d3r+.
E˜s(kx, ky, kz)=αkn2(2π)3G˜(k)E˜s(0)(k)s˜(k-k)d3k+α2kn2(2π)3G˜(k)E˜s(1)(k)s˜(k-k)d3k+ ,
G˜±(kx, ky, kz)=i2π2δ[(kn2-kx2-ky2)kz]±(kn2-kx2-ky2)1/2.
E˜s±(1)(kx, ky)=±ikn24π(kn2-kx2-ky2)1/2×E˜+(0)(kx, ky)·s˜[kx-kx, ky-ky, ±(kn2-kx2-ky2)1/2-(kn2-kx2-kyz2)1/2]dkxdky
E˜s±(2)(kx, ky)=kn4(4π)2(kn2-kx2-ky2)1/2×E˜+(0)(kx, kyz)(kn2-kx2-ky2)1/2·(s˜(k-k+)-s˜(k-k))dkxdky·s˜(k-k±)dkxdky,
k-k±=[kx-kx, ky-ky, (kn2-kx2-ky2)1/2(kn2-kx2-ky2)1/2]
s˜(kx, ky, kz)=3[sin(ka)-(ka)cos(ka)]/(ka)3V(a),
s(r-(bx, by, 0))exp(ikr)d3r=s˜(k)exp[i(kxbx+kyby)].
E˜s(kx, ky)=E˜sx(kx, ky)Dx(kx, ky)+E˜sz(kx, ky)Dz(kx, ky),
I˜s=ε0c|E˜sxDx+E˜szDz|2=ε0c(|E˜sxxDx+E˜sxzDz|2+|E˜szxDx+E˜szzDz|2)=ε0c|(αE˜s(1)+α2E˜s(2)+)xDx+(αE˜s(1)+α2E˜s(2)+)zDz|2.
Csca=1Imkn4-knkn-knkn[I˜s+(kx, ky)+I˜s-(kx, ky)]kn(kn2-kx2-ky2)1/2dkxdky=1Imkn202π0π/2Is+(θ, φ)sin(θ)dθdφ+1Imkn2×02ππ/2πIs-(θ, φ)sin(θ)dθdφ.
Qsca=Csca/Cgeo.
ksz=ktz-krz=1CscaImkn4-knkn-knknkz(kx, ky)[I˜s+(kx, ky)+I˜s-(kx, ky)]kn(kn2-kx2-ky2)1/2dkxdky.
Cpr,z=Cscagz1kn=Csca[kiz-(ktz-krz)]1kn,
g=(gxex+gyey+gzez),
Fscat=Pscagck0=nmcPscag/kn.
Fscat=nmcPsca(g/kn)=nmαΔMΔt.
F(r)=Vf(r)dV=Fgrad(r)+Fscat(r)=Vαnm2cIm(r)dV+nmcImCsca(g/kn).
Fgrad(r)=αnm2cVIm(r)dV=αnm2cVIm(r)nAdA,
Q(r)=F(r)cPnm=Qgrad(r)+Qscat(r).
Δϕ=2a/λ0(ns-nm)2π.
Δψ=d2π/λ(nm cos θm-ni cos θi).
Wz(x, 0, z)-Wz(x, 0, z0)=-z0z[Fgrad,z(x, 0, z)+Fscat,z(x, 0, z)]dz.
(2+kn2)G(r-r)=-δ(r-r).
(-k·k+kn2)G˜(k)=-1,
G˜(kx, ky, kz)=1|k|2-kn2=1kx2+ky2+kz2-kn2.
exp(iknR)4πR=1(2π)31kx2+ky2+kz2-kn2×exp[-i(kxx+kyy+kzz)]dkxdkydkz.
exp(iknR)4πR=i8π2exp[-iz|(kn2-kx2-ky2)1/2|]|(kn2-kx2-ky2)1/2|×exp[-i(kxx+kyy)]dkxdky.
1(2π)1kx2+ky2+kz2-kn2exp[-ikzz]dkz=i2exp[-iz|(kn2-kx2-ky2)1/2|]|(kn2-kx2-ky2)1/2|].
1kx2+ky2+kz2-kn2=i22πexp[-iz|(kn2-kx2-ky2)1/2|]|(kn2-kx2-ky2)1/2|]exp[ikzz]dz.
E˜s(0)(kx, ky, kz)=E0δ(kx, ky, kz-kn).
E˜s(1)(kx, ky, kz)=kn2(2π)3G˜(k)E0s˜(kx, ky, kz-kn),
E˜s±(1)(kx, ky)=±ikn24πkn2-k2E0s˜(kx, ky, ±kn2-k2-kn).
E˜s(2)(kx, ky, kz)=kn2(2π)32G˜(k)E0s˜(kx, ky, kz-kn)×G˜(k)s˜(k-k)d3k.
E˜s(2)(kx, ky, kz)=-kn4δ(|kn2-k2|-kz)(4π)2|kn2-k2|×E0δ(kn2-k2-kz)kn2-k2·[s˜(kx, ky, kz-kn)-s˜(kx, ky, -kz-kn)]·s˜(kx-kx, ky-ky, kz-kz)dkdkydkz.
E˜s±(2)(kx, ky)=kn4(4π)2kn2-k2E0kn2-k2·[s˜(kx, ky, kn2-k2-kn)-s˜(kx, ky, -kn2-k2-kn)]·s˜ (kx-kx, ky-ky, ±kn2-k2kn2-k2)dkxdky.

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