Abstract

A detailed study of time-averaged electromagnetic forces on subwavelength-sized particles is presented. An analytical decomposition of the force into gradient and scattering-plus-absorption components is carried out, on the basis of which the attractive or repulsive behavior of the force is explained. Small metallic particles are shown to experience both kinds of forces; which kind also depends on the excitation of surface plasmons. Resonances give rise to enhancements of both the scattering and the absorption forces, but the gradient force can become negligible. Also, close to resonant wavelengths, the gradient force can be maximum, while both the scattering and the absorption forces remain large. Comparisons of analytic results with rigorous calculations allow the establishment of ranges of validity of the dipolar approximation for these forces.

© 2003 Optical Society of America

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    [CrossRef] [PubMed]

2002 (5)

N. Calander, M. Willander, “Optical trapping of single fluorescent molecules at the detection spots of nanoprobes,” Phys. Rev. Lett. 89, 143603 (2002).
[CrossRef] [PubMed]

H. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

J. R. Arias-González, M. Nieto-Vesperinas, M. Lester, “Modeling photonic-force microscopy with metallic particles under plasmon eigenmode excitation,” Phys. Rev. B 65, 115402 (2002).
[CrossRef]

R. R. Agayan, F. Gittes, R. Kopelman, C. F. Schmidt, “Optical trapping near resonance absorption,” Appl. Opt. 41, 2318–2327 (2002).
[CrossRef] [PubMed]

J. R. Arias-González, M. Nieto-Vesperinas, “Radiation pressure over dielectric and metallic nanocylinders on surfaces: polarization dependence and plasmon resonance conditions,” Opt. Lett. 27, 2149–2151 (2002).
[CrossRef]

2001 (3)

2000 (4)

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

K. Sasaki, J. Hotta, K. Wada, H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25, 1385–1387 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric substrate,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

1999 (2)

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

J. R. Arias-González, M. Nieto-Vesperinas, A. Madrazo, “Morphology-dependent resonances in the scattering of electromagnetic waves from an object buried beneath a plane or a random rough surface,” J. Opt. Soc. Am. A 16, 2928–2934 (1999).
[CrossRef]

1998 (3)

A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, S71–S73 (1998).
[CrossRef]

J. Hotta, K. Sasaki, H. Masuhara, Y. Morishima, “Laser-controlled assembling of repulsive unimolecular micelles in aqueous solution,” J. Phys. Chem. B 102, 7687–7690 (1998).
[CrossRef]

H. Furukawa, I. Yamaguchi, “Optical trapping of metallic particles by a fixed Gaussian beam,” Opt. Lett. 23, 216–218 (1998).
[CrossRef]

1997 (2)

1996 (2)

1994 (1)

1992 (3)

S. Kawata, T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17, 772–774 (1992).
[CrossRef] [PubMed]

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

1991 (3)

1988 (1)

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

1986 (1)

1985 (1)

A. Ashkin, J. M. Dziedzic, “Observation of radiation-pressure trapping of particles by alternating light beams,” Phys. Rev. Lett. 54, 1245–1248 (1985).
[CrossRef] [PubMed]

1984 (1)

1980 (1)

A. Ashkin, “Applications of laser-radiation pressure,” Science 210, 1081–1088 (1980).
[CrossRef] [PubMed]

1973 (1)

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

Agayan, R. R.

Arias-González, J. R.

Ashkin, A.

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, J. M. Dziedzic, “Observation of radiation-pressure trapping of particles by alternating light beams,” Phys. Rev. Lett. 54, 1245–1248 (1985).
[CrossRef] [PubMed]

A. Ashkin, “Stable radiation-pressure particle traps using alternating light beams,” Opt. Lett. 9, 454–456 (1984).
[CrossRef] [PubMed]

A. Ashkin, “Applications of laser-radiation pressure,” Science 210, 1081–1088 (1980).
[CrossRef] [PubMed]

Bian, R. X.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

Bjorkholm, J. E.

Block, S. M.

Bohren, C. F.

C. E. Dungey, C. F. Bohren, “Light scattering by nonspherical particles: a refinement to the coupled-dipole method,” J. Opt. Soc. Am. A 8, 81–87 (1991).
[CrossRef]

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

Bustamante, C.

S. B. Smith, Y. Cui, C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-strandedand single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef] [PubMed]

Calander, N.

N. Calander, M. Willander, “Optical trapping of single fluorescent molecules at the detection spots of nanoprobes,” Phys. Rev. Lett. 89, 143603 (2002).
[CrossRef] [PubMed]

Chaumet, P. C.

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric substrate,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

Chu, S.

Cui, Y.

S. B. Smith, Y. Cui, C. Bustamante, “Overstretching B-DNA: the elastic response of individual double-strandedand single-stranded DNA molecules,” Science 271, 795–799 (1996).
[CrossRef] [PubMed]

Draine, B. T.

B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophys. J. 333, 848–872 (1988).
[CrossRef]

Dungey, C. E.

Dziedzic, J. M.

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, J. M. Dziedzic, “Observation of radiation-pressure trapping of particles by alternating light beams,” Phys. Rev. Lett. 54, 1245–1248 (1985).
[CrossRef] [PubMed]

Florin, E.-L.

A. Jonáš, P. Zemánek, E.-L. Florin, “Single-beam trapping in front of reflective surfaces,” Opt. Lett. 26, 1466–1468 (2001).
[CrossRef]

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [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, S71–S73 (1998).
[CrossRef]

Furukawa, H.

Gittes, F.

Gordon, J. P.

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

Hörber, J. K. H.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [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, S71–S73 (1998).
[CrossRef]

Hotta, J.

K. Sasaki, J. Hotta, K. Wada, H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25, 1385–1387 (2000).
[CrossRef]

J. Hotta, K. Sasaki, H. Masuhara, Y. Morishima, “Laser-controlled assembling of repulsive unimolecular micelles in aqueous solution,” J. Phys. Chem. B 102, 7687–7690 (1998).
[CrossRef]

Huffman, D. R.

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

Inouye, Y.

Jonáš, A.

Käll, M.

H. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

Kawata, S.

Kitamura, N.

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70, 3829–3836 (1991).
[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]

Kopelman, R.

Koshioka, M.

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70, 3829–3836 (1991).
[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]

Lester, M.

J. R. Arias-González, M. Nieto-Vesperinas, M. Lester, “Modeling photonic-force microscopy with metallic particles under plasmon eigenmode excitation,” Phys. Rev. B 65, 115402 (2002).
[CrossRef]

M. Lester, J. R. Arias-González, M. Nieto-Vesperinas, “Fundamentals and model of photonic-force microscopy,” Opt. Lett. 26, 707–709 (2001).
[CrossRef]

Madrazo, A.

Masuhara, H.

K. Sasaki, J. Hotta, K. Wada, H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25, 1385–1387 (2000).
[CrossRef]

J. Hotta, K. Sasaki, H. Masuhara, Y. Morishima, “Laser-controlled assembling of repulsive unimolecular micelles in aqueous solution,” J. Phys. Chem. B 102, 7687–7690 (1998).
[CrossRef]

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70, 3829–3836 (1991).
[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]

Misawa, H.

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70, 3829–3836 (1991).
[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]

Morishima, Y.

J. Hotta, K. Sasaki, H. Masuhara, Y. Morishima, “Laser-controlled assembling of repulsive unimolecular micelles in aqueous solution,” J. Phys. Chem. B 102, 7687–7690 (1998).
[CrossRef]

Nakamura, O.

Nieto-Vesperinas, M.

J. R. Arias-González, M. Nieto-Vesperinas, “Radiation pressure over dielectric and metallic nanocylinders on surfaces: polarization dependence and plasmon resonance conditions,” Opt. Lett. 27, 2149–2151 (2002).
[CrossRef]

J. R. Arias-González, M. Nieto-Vesperinas, M. Lester, “Modeling photonic-force microscopy with metallic particles under plasmon eigenmode excitation,” Phys. Rev. B 65, 115402 (2002).
[CrossRef]

M. Lester, J. R. Arias-González, M. Nieto-Vesperinas, “Fundamentals and model of photonic-force microscopy,” Opt. Lett. 26, 707–709 (2001).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. 25, 1065–1067 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric substrate,” Phys. Rev. B 62, 11185–11191 (2000).
[CrossRef]

P. C. Chaumet, M. Nieto-Vesperinas, “Coupled dipole method determination of the electromagnetic force on a particle over a flat dielectric substrate,” Phys. Rev. B 61, 14119–14127 (2000).
[CrossRef]

J. R. Arias-González, M. Nieto-Vesperinas, A. Madrazo, “Morphology-dependent resonances in the scattering of electromagnetic waves from an object buried beneath a plane or a random rough surface,” J. Opt. Soc. Am. A 16, 2928–2934 (1999).
[CrossRef]

A. Madrazo, M. Nieto-Vesperinas, “Surface structure and polariton interactions in the scattering of electromagnetic waves from a cylinder in front of a conducting grating: theory for the reflection photon scanning tunneling microscope,” J. Opt. Soc. Am. A 13, 785–795 (1996).
[CrossRef]

Novotny, L.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

Okada, T.

Pralle, A.

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
[CrossRef] [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, S71–S73 (1998).
[CrossRef]

Prummer, M.

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

Raether, H.

H. Raether, Surface Plasmons (Springer-Verlag, Berlin, 1988).

Rohrbach, A.

Sasaki, K.

K. Sasaki, J. Hotta, K. Wada, H. Masuhara, “Analysis of radiation pressure exerted on a metallic particle within an evanescent field,” Opt. Lett. 25, 1385–1387 (2000).
[CrossRef]

J. Hotta, K. Sasaki, H. Masuhara, Y. Morishima, “Laser-controlled assembling of repulsive unimolecular micelles in aqueous solution,” J. Phys. Chem. B 102, 7687–7690 (1998).
[CrossRef]

H. Misawa, K. Sasaki, M. Koshioka, N. Kitamura, H. Masuhara, “Multibeam laser manipulation and fixation of microparticles,” Appl. Phys. Lett. 60, 310–312 (1992).
[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[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]

H. Misawa, M. Koshioka, K. Sasaki, N. Kitamura, H. Masuhara, “Three-dimensional optical trapping and laser ablation of a single polymer latex particle in water,” J. Appl. Phys. 70, 3829–3836 (1991).
[CrossRef]

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Smith, S. B.

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A. Rohrbach, E. H. K. Stelzer, “Optical trapping of di-electric particles in arbitrary fields,” J. Opt. Soc. Am. A 18, 839–853 (2001).
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A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light,” Microsc. Res. Tech. 44, 378–386 (1999).
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A. Pralle, E.-L. Florin, E. H. K. Stelzer, J. K. H. Hörber, “Local viscosity probed by photonic force microscopy,” Appl. Phys. A 66, S71–S73 (1998).
[CrossRef]

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N. Calander, M. Willander, “Optical trapping of single fluorescent molecules at the detection spots of nanoprobes,” Phys. Rev. Lett. 89, 143603 (2002).
[CrossRef] [PubMed]

Xie, X. S.

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

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H. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

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Appl. Opt. (1)

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, S71–S73 (1998).
[CrossRef]

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[CrossRef]

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Masuhara, “Optical trapping of a metal particle and a water droplet by a scanning laser beam,” Appl. Phys. Lett. 60, 807–809 (1992).
[CrossRef]

Astrophys. J. (1)

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[CrossRef]

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[CrossRef]

J. Opt. Soc. Am. A (4)

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[CrossRef]

Microsc. Res. Tech. (1)

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[CrossRef]

Phys. Rev. Lett. (4)

L. Novotny, R. X. Bian, X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. 79, 645–648 (1997).
[CrossRef]

A. Ashkin, J. M. Dziedzic, “Observation of radiation-pressure trapping of particles by alternating light beams,” Phys. Rev. Lett. 54, 1245–1248 (1985).
[CrossRef] [PubMed]

N. Calander, M. Willander, “Optical trapping of single fluorescent molecules at the detection spots of nanoprobes,” Phys. Rev. Lett. 89, 143603 (2002).
[CrossRef] [PubMed]

H. Xu, M. Käll, “Surface-plasmon-enhanced optical forces in silver nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Geometry of the system.

Fig. 2
Fig. 2

(a) Real and (b) imaginary parts of the dielectric permittivity for silver in vacuum (black thick curves) and in water (black thin curves) and gold in vacuum (gray thick curves) and in water (gray thin curves). The top figure shows the force on a gold cylinder (a=10 nm) in vacuum (thick curve) and in water (thin curve) exactly calculated by means of the partial-wave series. The same calculation is performed via the dipole approximation: a gold cylinder in vacuum (thin curve with inverted triangles) and in water (thin curve with up-pointing triangles). The illumination is done with a propagating plane wave.

Fig. 3
Fig. 3

Dipole approximation. (From left to right) top: real part, imaginary part, and modulus of the polarizability for a silver sphere (a=10 nm) in vacuum. Middle: vertical component, horizontal component, and modulus of the force on the same sphere (d=20 nm) under evanescent plane-wave incidence (θ0=50°>θc=41.47°). Bottom: horizontal component, vertical component, and modulus of the force on the same sphere under Gaussian beam illumination (W=6000 nm, θ0=0) at x0=-200 nm. Plain curves, Clausius–Mossotti polarizability with the radiative-reaction term; curves with symbol, polarizability of Dungey and Bohren.

Fig. 4
Fig. 4

Dipole approximation. (From left to right) top: real part, imaginary part, and modulus of the P-wave polarizability (Clausius–Mossotti equation with the radiative-reaction term) for a gold cylinder (a=10 nm) in water. Middle: vertical component, horizontal component, and modulus of the force on this cylinder in water under P-polarized evanescent plane-wave incidence at d=20 nm. Solid curves, θ0=62.5°; dashed curves, θ0=70°. θc=61.98°. Bottom: horizontal component, vertical component, and modulus of the force on this cylinder in water under P-polarized Gaussian beam incidence (θ0=0) at x0=20 nm. Solid curves, W=6000 nm; dashed curves, W=10,000 nm.

Fig. 5
Fig. 5

ET calculation of the potential energy of a gold cylinder (a=10 nm) immersed in water under the influence of a P-polarized Gaussian beam with axis parallel to the cylinder axis along the line x=0 (θ0=0). The cylinder moves along OX. Black thick curve, λ = 500 nm and W=6000 nm; black thin curve, λ = 479 nm and W=6000 nm; dashed curve, λ = 1064 nm and W=6000 nm; gray thick curve, λ = 479 nm and W=10,000 nm. Curves with crosses denote the same calculations performed with the dipole approximation. kB is the Boltzmann constant, and T=293 K is the temperature.

Fig. 6
Fig. 6

ET calculation of the potential energy of an isolated cylinder (a=10 nm) immersed in water under the influence of a P-polarized evanescent wave (plane of incidence, OXZ). The cylinder moves along the OZ axis. (a) Gold, λ=479 nm; (b) gold, λ=500 nm (on resonance); (c) gold, λ=1064 nm; (d) glass, λ=632.8 nm. Plain lines, ET calculation; symbols, dipole approximation. Black curves and crosses, θ0=62.5°; gray curves and inverse triangles, θ0=65°; dashed curves and triangles, θ0=70°. θc=61.98°. kB is the Boltzmann constant, and T=293 K is the temperature.

Fig. 7
Fig. 7

ET calculation of the potential energy of an isolated cylinder immersed in water under the influence of a P-polarized evanescent wave with θ0=62.5° (>θc=61.98°) (plane of incidence, OXZ). The cylinder moves along the OZ axis. (a) Gold, λ=500 nm (on resonance); (b) gold, λ=1064 nm; (c) glass, λ=632.8 nm. Plain lines, ET calculation; symbols, dipole approximation. Black curves and crosses, a=20 nm; gray curves and inverse triangles, a=30 nm; dashed curves and triangles, a=50 nm; dotted curves and circles, a=70 nm; dotted–dashed curves and squares, a=100 nm. kB is the Boltzmann constant, and T=293 K is the temperature.

Fig. 8
Fig. 8

Relative difference between the normalized force [F/exp(-2qz), see the text for details] calculated from the ET and from the dipole approximation. A cylinder (a=10 nm) is immersed in water and illuminated by a P-polarized evanescent wave (plane of incidence, OXZ). (a), (b) Error in Fx; (c), (d) error in Fz. (a), (c) Error as a function of the decay length of the evanescent wave; (b), (d) error as a function of the angle of incidence. Black curves: gold, λ=479 nm; gray curves: gold, λ=500 nm (on resonance); dashed curves: gold, λ=1064 nm; dotted curves: glass, λ=632.8 nm.

Equations (24)

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E(inc)(r, t)=[0, ΦS(inc)(r), 0]exp(-iωt),
H(inc)(r, t)=[0, ΦP(inc)(r), 0]exp(-iωt),
Φγ(inc)(r)=exp[i1k0(x sin θ0+z cos θ0)g(x, z)]×exp[-(x cos θ0-z sin θ0)2/W2],
g(x, z)=1+11k02W22W2 (x cos θ0-z sin θ0)2-1.
F=18πReΣd2r[E(r, ω)n]E*(r, ω)+[μH(r, ω)n]H*(r, ω)-12 [|E(r, ω)|2+μ|H(r, ω)|2]n,
Fi(r)=12ReαEj(r) Ej*(r)xi,
α(ω)=α0(ω)1-23ik3α0(ω),
E(r)=E0(r)exp(ikr).
F=14Re{α}|E0|2+12kIm{α}|E0|2-12Im{α}Im{E0E0*}.
E=(0, 1, 0)Texp(iKx)exp(-qz)
E=(-iq, 0, K) Tkexp(iKx)exp(-qz)
Fx=|T|22 K Im{α}exp(-2qz),
Fz=-|T|22 q Re{α}exp(-2qz).
Fx=|T|28πKkexp(-2qz)Cext,
Cext=4πka3Im-1+2+8π3 k4a6-1+22.
Fx=-|T|2xW2Re{α},
Fz=|T|22 k Im{α},
Re{α0}=a3(-1)(+2)+2(+2)2+2,
Im{α0}=a33(+2)2+2,
α(p)(ω)=α0(p)(ω)1-ik2πα0(p)(ω)/2,
α0(p)(ω)=a22(ω)-1(ω)+1.
V(x)=|T|2x22W2Re{α}.
V(z)=|T|24Re{α}[exp(-2qa)-exp(-2qz)].
V(z)=|T|22Re{α}q(z-a),

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