Abstract

We present an efficient, discrete-dipole-approximation-based method for computing gradient and nongradient contributions to the optically induced force on neutral, polarizable particles in a field. We compare numerical data from this method with those generated using previously devised computational approaches for computing total forces. The agreement is generally adequate, and rounding error is the likely cause for differences among results obtained from the three methods. For both one- and two-sphere targets, nongradient forces generally make a nonnegligible contribution. For spheres, the gradient force often nearly cancels a component of the nongradient force, so that the radiation-pressure component is approximately equal to the net force. These results are contrary to the commonly assumed dominance of the gradient force for nanometer-sized particles.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Ashkin and J. M. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971).
    [CrossRef]
  2. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  3. E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
    [CrossRef] [PubMed]
  4. P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
    [CrossRef]
  5. A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
    [CrossRef]
  6. S. B. Smith, Y. J. Cui, and C. Bustamante, "Overstretching B-DNA: the elastic response of individual double-stranded and single-stranded DNA molecules," Science 271, 795-799 (1996).
    [CrossRef] [PubMed]
  7. K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
    [CrossRef] [PubMed]
  8. T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
    [CrossRef] [PubMed]
  9. M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
    [CrossRef] [PubMed]
  10. M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
    [CrossRef]
  11. S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
    [CrossRef] [PubMed]
  12. J. R. Arias-Gonzalez, M. Nieto-Vesperinas, and M. Lester, "Modeling photonic force microscopy with metallic particles under plasmon eigenmode excitation," Phys. Rev. B 65, 115402 (2002).
    [CrossRef]
  13. T. Sugiura, "Laser trapping of a metallic probe for near field microscopy," Top. Appl. Phys. 81, 143-161 (2001).
    [CrossRef]
  14. L. Malmqvist and H. M. Hertz, "Trapped particle optical microscopy," Opt. Commun. 94, 19-24 (1992).
    [CrossRef]
  15. P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
    [CrossRef]
  16. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
    [CrossRef]
  17. C. Dominik and A. G. G. M. Tielens, "The physics of dust coagulation and the structure of dust aggregates in space," Astrophys. J. 480, 647-673 (1997).
    [CrossRef]
  18. A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
    [CrossRef] [PubMed]
  19. R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
    [CrossRef] [PubMed]
  20. M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
    [CrossRef]
  21. A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
    [CrossRef]
  22. R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
    [CrossRef] [PubMed]
  23. A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
    [CrossRef] [PubMed]
  24. P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422 (2001).
    [CrossRef]
  25. L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
    [CrossRef]
  26. V. Wong and M. A. Ratner, "On the size dependence of gradient and non-gradient optical forces in silver nanoparticles" (submitted to J. Opt. Soc. Am. B).
  27. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, "Radiation forces in the discrete-dipole approximation," J. Opt. Soc. Am. A 18, 1944-1953 (2001).
    [CrossRef]
  28. P. C. Chaumet and M. Nieto-Vesperinas, "Time-averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000).
    [CrossRef]
  29. F. J. Garcia de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transf. 89, 3-9 (2004).
    [CrossRef]
  30. T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
    [CrossRef]
  31. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  32. B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
    [CrossRef]
  33. S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
    [CrossRef]
  34. L. Novotny, "Forces in optical near-fields," Top. Appl. Phys. 81, 123-141 (2001).
    [CrossRef]
  35. J. R. Arias-Gonzalez and M. Nieto-Vesperinas, "Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions," J. Opt. Soc. Am. A 20, 1201-1209 (2003).
    [CrossRef]
  36. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
    [CrossRef]
  37. G. E. Hay, Vector and Tensor Analysis (Dover, 1953).
  38. V. Wong and M. A. Ratner, "Gradient and nongradient contributions to plasmon enhanced optical forces on silver nanoparticles," Phys. Rev. B 73, 075416 (2006).
    [CrossRef]
  39. A. Ashkin and J. P. Gordon, "Stability of radiation-pressure particle traps: an optical Earnshaw theorem," Opt. Lett. 8, 511-513 (1983).
    [CrossRef] [PubMed]
  40. B. T. Draine and P. J. Flatau, "User Guide to the Discrete Dipole Approximation code DDSCAT 6.0" arxiv.org e-print archive, astro-ph/0300969, 2003, http://arxiv.org/abs/astro-ph/0300969.
  41. W. A. Kraus and G. C. Schatz, "Plasmon resonance broadening in small metal particles," J. Chem. Phys. 79, 6130-6139 (1983).
    [CrossRef]
  42. R. Fuchs and F. Claro, "Multipolar response of small metallic spheres: nonlocal theory," Phys. Rev. B 35, 3722-3727 (1987).
    [CrossRef]
  43. S. D. Druger and B. V. Bronk, "Internal and scattered electric fields in the discrete dipole approximation," J. Opt. Soc. Am. B 16, 2239-2246 (1999).
    [CrossRef]
  44. A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
    [CrossRef]
  45. M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
    [CrossRef]
  46. P. C. Chaumet and 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]
  47. J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

2006 (1)

V. Wong and M. A. Ratner, "Gradient and nongradient contributions to plasmon enhanced optical forces on silver nanoparticles," Phys. Rev. B 73, 075416 (2006).
[CrossRef]

2005 (3)

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
[CrossRef]

A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
[CrossRef] [PubMed]

2004 (5)

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

F. J. Garcia de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transf. 89, 3-9 (2004).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
[CrossRef]

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

2003 (5)

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, "Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions," J. Opt. Soc. Am. A 20, 1201-1209 (2003).
[CrossRef]

M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
[CrossRef]

A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
[CrossRef]

2002 (4)

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

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

2001 (7)

T. Sugiura, "Laser trapping of a metallic probe for near field microscopy," Top. Appl. Phys. 81, 143-161 (2001).
[CrossRef]

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

L. Novotny, "Forces in optical near-fields," Top. Appl. Phys. 81, 123-141 (2001).
[CrossRef]

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422 (2001).
[CrossRef]

A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, "Radiation forces in the discrete-dipole approximation," J. Opt. Soc. Am. A 18, 1944-1953 (2001).
[CrossRef]

2000 (2)

P. C. Chaumet and 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 and 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)

K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

S. D. Druger and B. V. Bronk, "Internal and scattered electric fields in the discrete dipole approximation," J. Opt. Soc. Am. B 16, 2239-2246 (1999).
[CrossRef]

1997 (3)

L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

C. Dominik and A. G. G. M. Tielens, "The physics of dust coagulation and the structure of dust aggregates in space," Astrophys. J. 480, 647-673 (1997).
[CrossRef]

1996 (2)

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

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

1994 (1)

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

1992 (1)

L. Malmqvist and H. M. Hertz, "Trapped particle optical microscopy," Opt. Commun. 94, 19-24 (1992).
[CrossRef]

1988 (1)

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

1987 (1)

R. Fuchs and F. Claro, "Multipolar response of small metallic spheres: nonlocal theory," Phys. Rev. B 35, 3722-3727 (1987).
[CrossRef]

1986 (1)

S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
[CrossRef]

1983 (2)

A. Ashkin and J. P. Gordon, "Stability of radiation-pressure particle traps: an optical Earnshaw theorem," Opt. Lett. 8, 511-513 (1983).
[CrossRef] [PubMed]

W. A. Kraus and G. C. Schatz, "Plasmon resonance broadening in small metal particles," J. Chem. Phys. 79, 6130-6139 (1983).
[CrossRef]

1971 (1)

A. Ashkin and J. M. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

1970 (1)

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

Arias-Gonzalez, J. R.

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, "Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions," J. Opt. Soc. Am. A 20, 1201-1209 (2003).
[CrossRef]

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

Ashkin, A.

A. Ashkin and J. P. Gordon, "Stability of radiation-pressure particle traps: an optical Earnshaw theorem," Opt. Lett. 8, 511-513 (1983).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

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

Bachelot, R.

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

Barchiesi, D.

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

Betz, T.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Bian, X.

L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Bishop, A. I.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Bjerneld, E. J.

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

Block, S. M.

K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

Bronk, B. V.

Brus, L.

M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
[CrossRef]

Brus, L. E.

A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
[CrossRef] [PubMed]

Bryant, G. W.

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

Bustamante, C.

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

Butler, W. F.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Callegari, A.

A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
[CrossRef]

Cao, Y. C.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Chaumet, P. C.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
[CrossRef]

P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422 (2001).
[CrossRef]

P. C. Chaumet and 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 and 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]

Chergui, M.

A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
[CrossRef]

Chu, S.

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

Claro, F.

R. Fuchs and F. Claro, "Multipolar response of small metallic spheres: nonlocal theory," Phys. Rev. B 35, 3722-3727 (1987).
[CrossRef]

Collinge, M. J.

Cui, Y. J.

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

Dees, B.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Dholakia, K.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

Dominik, C.

C. Dominik and A. G. G. M. Tielens, "The physics of dust coagulation and the structure of dust aggregates in space," Astrophys. J. 480, 647-673 (1997).
[CrossRef]

Draine, B. T.

M. J. Collinge and B. T. Draine, "Discrete-dipole approximation with polarizabilities that account for both finite wavelength and target geometry," J. Opt. Soc. Am. A 21, 2023-2028 (2004).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

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

B. T. Draine and P. J. Flatau, "User Guide to the Discrete Dipole Approximation code DDSCAT 6.0" arxiv.org e-print archive, astro-ph/0300969, 2003, http://arxiv.org/abs/astro-ph/0300969.

Druger, S. D.

Dziedzic, J. M.

A. Ashkin and J. M. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

Ehrlicher, A.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Elbaum, M.

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

Flatau, P. J.

B. T. Draine and P. J. Flatau, "User Guide to the Discrete Dipole Approximation code DDSCAT 6.0" arxiv.org e-print archive, astro-ph/0300969, 2003, http://arxiv.org/abs/astro-ph/0300969.

Forster, A. H.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Frijlink, M.

Fuchs, R.

R. Fuchs and F. Claro, "Multipolar response of small metallic spheres: nonlocal theory," Phys. Rev. B 35, 3722-3727 (1987).
[CrossRef]

Fytas, G.

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

Garcia de Abajo, F. J.

F. J. Garcia de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transf. 89, 3-9 (2004).
[CrossRef]

Gauthier, R. C.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Gordon, J. P.

Grover, C. P.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Grover, S. C.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Hadjichristidis, N.

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

Hagen, N.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Hallock, A. J.

A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
[CrossRef] [PubMed]

Hao, E.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Hay, G. E.

G. E. Hay, Vector and Tensor Analysis (Dover, 1953).

Heckenberg, N. R.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Hertz, H. M.

L. Malmqvist and H. M. Hertz, "Trapped particle optical microscopy," Opt. Commun. 94, 19-24 (1992).
[CrossRef]

Hoekstra, A. G.

Huang, P.

M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
[CrossRef]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

Jin, R.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Kall, M.

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

Kariv, I.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Kas, J.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Klein, E.

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

Koch, D.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Kraus, W. A.

W. A. Kraus and G. C. Schatz, "Plasmon resonance broadening in small metal particles," J. Chem. Phys. 79, 6130-6139 (1983).
[CrossRef]

Lachish-Zalait, A.

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

Lerondel, G.

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

Lester, M.

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

MacDonald, M. P.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

Maillard, M.

M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
[CrossRef]

Malmqvist, L.

L. Malmqvist and H. M. Hertz, "Trapped particle optical microscopy," Opt. Commun. 94, 19-24 (1992).
[CrossRef]

Marchand, P. J.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Mercer, E. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Metraux, G. S.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Milner, V.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Mirkin, C. A.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Murty, K. V. G. K.

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

Nieminen, T. A.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Nieto-Vesperinas, M.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
[CrossRef]

J. R. Arias-Gonzalez and M. Nieto-Vesperinas, "Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions," J. Opt. Soc. Am. A 20, 1201-1209 (2003).
[CrossRef]

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

P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422 (2001).
[CrossRef]

P. C. Chaumet and 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 and 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]

Novotny, L.

L. Novotny, "Forces in optical near-fields," Top. Appl. Phys. 81, 123-141 (2001).
[CrossRef]

L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Perkins, T. T.

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

Pispas, S.

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

Prikulis, J.

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

Quake, S. R.

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

Rahmani, A.

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
[CrossRef]

Raizen, M. G.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Ratner, M. A.

V. Wong and M. A. Ratner, "Gradient and nongradient contributions to plasmon enhanced optical forces on silver nanoparticles," Phys. Rev. B 73, 075416 (2006).
[CrossRef]

V. Wong and M. A. Ratner, "On the size dependence of gradient and non-gradient optical forces in silver nanoparticles" (submitted to J. Opt. Soc. Am. B).

Raymond, D. E.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Redmond, P. L.

A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
[CrossRef] [PubMed]

Royer, P.

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

Rubinsztein-Dunlop, H.

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

Schatz, G. C.

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

W. A. Kraus and G. C. Schatz, "Plasmon resonance broadening in small metal particles," J. Chem. Phys. 79, 6130-6139 (1983).
[CrossRef]

Schnitzer, M. J.

K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

Sigel, R.

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

Skirtach, A. G.

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

Sloot, P. M. A.

Smith, D. E.

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

Smith, S. B.

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

Spalding, G. C.

M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

Stenholm, S.

S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
[CrossRef]

Stuhrmann, B.

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Sugiura, T.

T. Sugiura, "Laser trapping of a metallic probe for near field microscopy," Top. Appl. Phys. 81, 143-161 (2001).
[CrossRef]

Tielens, A. G. G. M.

C. Dominik and A. G. G. M. Tielens, "The physics of dust coagulation and the structure of dust aggregates in space," Astrophys. J. 480, 647-673 (1997).
[CrossRef]

Tonti, D.

A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
[CrossRef]

Tu, E.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Vainos, N.

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

Visscher, K.

K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

Wang, M. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Waters, L. B. F. M.

Weingartner, J. C.

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

Wong, V.

V. Wong and M. A. Ratner, "Gradient and nongradient contributions to plasmon enhanced optical forces on silver nanoparticles," Phys. Rev. B 73, 075416 (2006).
[CrossRef]

V. Wong and M. A. Ratner, "On the size dependence of gradient and non-gradient optical forces in silver nanoparticles" (submitted to J. Opt. Soc. Am. B).

Xie, X. S.

L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

Yang, J. M.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Zbaida, D.

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

Zhang, H. C.

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Adv. Funct. Mater. (1)

A. Lachish-Zalait, D. Zbaida, E. Klein, and M. Elbaum, "Direct surface patterning from solutions: localized microchemistry using a focused laser," Adv. Funct. Mater. 11, 218-223 (2001).
[CrossRef]

Appl. Phys. Lett. (1)

A. Ashkin and J. M. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283-285 (1971).
[CrossRef]

Astrophys. J. (5)

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains II. Grain alignment," Astrophys. J. 480, 633-646 (1997).
[CrossRef]

C. Dominik and A. G. G. M. Tielens, "The physics of dust coagulation and the structure of dust aggregates in space," Astrophys. J. 480, 647-673 (1997).
[CrossRef]

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

B. T. Draine and J. C. Weingartner, "Radiative torques on interstellar grains. I. Superthermal spin-up," Astrophys. J. 470, 551-565 (1996).
[CrossRef]

A. Rahmani, P. C. Chaumet, and G. W. Bryant, "On the importance of local-field corrections for polarizable particles on a finite lattice: application to the discrete dipole approximation," Astrophys. J. 607, 873-878 (2004).
[CrossRef]

ChemPhysChem (1)

E. J. Bjerneld, K. V. G. K. Murty, J. Prikulis, and M. Kall, "Laser-induced growth of Ag nanoparticles from aqueous solutions," ChemPhysChem 3, 116-119 (2002).
[CrossRef] [PubMed]

Comput. Phys. Commun. (1)

T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, "Numerical modelling of optical trapping," Comput. Phys. Commun. 142, 468-471 (2001).
[CrossRef]

J. Biomed. Opt. (1)

S. C. Grover, A. G. Skirtach, R. C. Gauthier, and C. P. Grover, "Automated single-cell sorting system based on optical trapping," J. Biomed. Opt. 6, 14-22 (2001).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

W. A. Kraus and G. C. Schatz, "Plasmon resonance broadening in small metal particles," J. Chem. Phys. 79, 6130-6139 (1983).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

F. J. Garcia de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transf. 89, 3-9 (2004).
[CrossRef]

Nano Lett. (2)

M. Maillard, P. Huang, and L. Brus, "Silver nanodisk growth by surface plasmon enhanced photoreduction of adsorbed [Ag+]," Nano Lett. 3, 1611-1615 (2003).
[CrossRef]

A. Callegari, D. Tonti, and M. Chergui, "Photochemically grown silver nanoparticles with wavelength-controlled size and shape," Nano Lett. 3, 1565-1568 (2003).
[CrossRef]

Nat. Biotechnol. (1)

M. M. Wang, E. Tu, D. E. Raymond, J. M. Yang, H. C. Zhang, N. Hagen, B. Dees, E. M. Mercer, A. H. Forster, I. Kariv, P. J. Marchand, and W. F. Butler, "Microfluidic sorting of mammalian cells by optical force switching," Nat. Biotechnol. 23, 83-87 (2005).
[CrossRef]

Nature (3)

M. P. MacDonald, G. C. Spalding, and K. Dholakia, "Microfluidic sorting in an optical lattice," Nature 426, 421-424 (2003).
[CrossRef] [PubMed]

K. Visscher, M. J. Schnitzer, and S. M. Block, "Single kinesin molecules studied with a molecular force clamp," Nature 400, 184-189 (1999).
[CrossRef] [PubMed]

R. Jin, Y. C. Cao, E. Hao, G. S. Metraux, G. C. Schatz, and C. A. Mirkin, "Controlling anisotropic nanoparticle growth through plasmon excitation," Nature 425, 487-490 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

L. Malmqvist and H. M. Hertz, "Trapped particle optical microscopy," Opt. Commun. 94, 19-24 (1992).
[CrossRef]

Opt. Lett. (2)

Philos. Trans. R. Soc. London Ser. A (2)

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, "Near-field photonic forces," Philos. Trans. R. Soc. London Ser. A 362, 719-737 (2004).
[CrossRef]

P. Royer, D. Barchiesi, G. Lerondel, and R. Bachelot, "Near-field optical patterning and structuring based on local-field enhancement at the extremity of a metal tip," Philos. Trans. R. Soc. London Ser. A 362, 821-842 (2004).
[CrossRef]

Phys. Rev. B (6)

P. C. Chaumet, A. Rahmani, and M. Nieto-Vesperinas, "Photonic force spectroscopy on metallic and absorbing nanoparticles," Phys. Rev. B 71, 045425 (2005).
[CrossRef]

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

P. C. Chaumet and M. Nieto-Vesperinas, "Optical binding of particles with or without the presence of a flat dielectric surface," Phys. Rev. B 64, 035422 (2001).
[CrossRef]

R. Fuchs and F. Claro, "Multipolar response of small metallic spheres: nonlocal theory," Phys. Rev. B 35, 3722-3727 (1987).
[CrossRef]

P. C. Chaumet and 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]

V. Wong and M. A. Ratner, "Gradient and nongradient contributions to plasmon enhanced optical forces on silver nanoparticles," Phys. Rev. B 73, 075416 (2006).
[CrossRef]

Phys. Rev. Lett. (2)

L. Novotny, X. Bian, and X. S. Xie, "Theory of nanometric optical tweezers," Phys. Rev. Lett. 79, 645-648 (1997).
[CrossRef]

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. (2)

A. J. Hallock, P. L. Redmond, and L. E. Brus, "Optical forces between metallic particles," Proc. Natl. Acad. Sci. U.S.A. 102, 1280-1284 (2005).
[CrossRef] [PubMed]

A. Ehrlicher, T. Betz, B. Stuhrmann, D. Koch, V. Milner, M. G. Raizen, and J. Kas, "Guiding neuronal growth with light," Proc. Natl. Acad. Sci. U.S.A. 99, 16024-16028 (2002).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
[CrossRef]

Science (3)

R. Sigel, G. Fytas, N. Vainos, S. Pispas, and N. Hadjichristidis, "Pattern formation in homogeneous polymer solutions induced by a continuous-wave visible laser," Science 297, 67-69 (2002).
[CrossRef] [PubMed]

T. T. Perkins, S. R. Quake, D. E. Smith, and S. Chu, "Relaxation of a single DNA molecule observed by optical microscopy," Science 264, 822-826 (1994).
[CrossRef] [PubMed]

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

Top. Appl. Phys. (2)

T. Sugiura, "Laser trapping of a metallic probe for near field microscopy," Top. Appl. Phys. 81, 143-161 (2001).
[CrossRef]

L. Novotny, "Forces in optical near-fields," Top. Appl. Phys. 81, 123-141 (2001).
[CrossRef]

Other (4)

G. E. Hay, Vector and Tensor Analysis (Dover, 1953).

V. Wong and M. A. Ratner, "On the size dependence of gradient and non-gradient optical forces in silver nanoparticles" (submitted to J. Opt. Soc. Am. B).

J. D. Jackson, Classical Electrodynamics (Wiley, 1975).

B. T. Draine and P. J. Flatau, "User Guide to the Discrete Dipole Approximation code DDSCAT 6.0" arxiv.org e-print archive, astro-ph/0300969, 2003, http://arxiv.org/abs/astro-ph/0300969.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Basic scheme for computing forces on particles. Forces on individual dipoles shown in (a) are summed to give total forces on each sphere shown in (b).

Fig. 2
Fig. 2

Results for a single sphere of radius A = 5 nm . (a) Target and field geometry. (b) Comparison of total force along k for the DW, HFWS, and present methods. The key also applies to (c). (c) Relative error for data in (b) using the DW result as the standard. (d) Contributions to force on the sphere.

Fig. 3
Fig. 3

Results for a two-sphere target. The large sphere has A = 5 nm and the small sphere has radius a = 1.08 nm . (a) Target and field geometry. (b) Total force on a two-sphere target along k for the DW, HFWS, and present methods at three intersphere distances. The key also applies to (c). (c) Relative error for data in (b) using the DW result as the standard. (d) Total force along E for each sphere. Large- and small-sphere data were multiplied by M = 1 , 5, 10 at d = 10.3 , 18.9, and 27.4 nm . Note that the spheres are always pushed toward each other in this configuration, (e) Relative error for data in (d) using HFWS as the standard. Large- and small-sphere data were multiplied by M = 80 , 8, 1 at d = 10.3 , 18.9, and 27.4 nm . (f) Contributions to the force along E for each sphere. Large-sphere data were multiplied by M = 1 , 2, 5 at d = 10.3 , 18.9, and 27.4 nm . Small-sphere data were multiplied by m = 1 , 10, 25.

Fig. 4
Fig. 4

Results for a two-sphere target. (a) Target and field geometry. (b) Total force on the two-sphere target along k for the DW, HFWS, and present methods. The key also applies to (c). (c) Relative error for data in (b) using the DW result as the standard. (d) Total force along k for each sphere. (e) Relative error for data in (d) using HFWS as the standard. (f) Contributions to force along k for each sphere. Small-sphere data were multiplied by m = 1 , 7.5, 15 at d = 10.3 , 18.9, 27.4 nm .

Fig. 5
Fig. 5

Results for a two-sphere target. (a) Target and field geometry. (b) Total force on the two-sphere target along k for the DW, HFWS, and present methods. The key also applies to (c). (c) Relative error for data in (b) using the DW results as the standard. (d) Total force along k for each sphere. (e) Relative error for data in (d) using HFWS as the standard. (f) Contributions to force along k for each sphere. Small-sphere data were multiplied by m = 1 , 17, 35 at d = 10.3 , l8.9, 27.4 nm .

Fig. 6
Fig. 6

Dependence of force contributions on the number of dipoles, N, used to represent a single sphere. Forces are along k, and λ = 402 nm ; (a) 10 nm radius; (b) 1 nm radius; (c) 0.1 nm radius.

Fig. 7
Fig. 7

Dependence of force contributions on the number of dipoles, N, used to represent a single prolate. Major to minor axis ratio, R = 3.2 . Forces are along k, λ = 670 nm , and E lies along the major axis. (a) Effective radius (see Ref. [31]), a = 10 nm , (b) a = 1 nm , (c) a = 0.1 nm .

Equations (87)

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

F ( t ) = Re [ p ( t ) ] Re [ E ( t ) ] + 1 c Re [ p ̇ ( t ) ] × Re [ B ( t ) ] ,
F = 1 2 Re [ ( p * ) E + i k p * × B ] ,
F = 1 2 Re [ α * ( E * ) E + i k α * ( E * × B ) ] ,
F = 1 2 { Re ( α ) Re [ ( E * ) E ] + Im ( α ) Im [ ( E * ) E ] + k Im ( α ) Re ( E * × B ) k Re ( α ) Im ( E * × B ) } .
( a b ) = ( a ) b + ( b ) a + a × ( × b ) + b × ( × a ) ,
Re [ ( E * ) E ] = 1 2 E 2 + k Im ( E * × B ) .
F = 1 4 Re ( α ) E 2 + k 2 Im ( α ) Re ( E * × B ) + 1 2 Im ( α ) Im [ ( E * ) E ] .
E inc = E 0 exp ( i k r i ) ,
E i j = exp ( i k r i j ) [ ( k 2 r i j + i k r i j 2 1 r i j 3 ) p j + ( k 2 r i j 3 i k r i j 2 + 3 r i j 3 ) n ̂ i j ( n ̂ i j p j ) ] ,
( E i * i ) E i = [ ( E inc * + k i E i k ) * i ] ( E inc + j i E i j )
( E i * i ) E i = ( E inc * i ) E inc + j i ( E inc * ) E i j + i i ( E i j * i ) E inc + k i j i ( E i k * i ) E i j .
( E inc * i ) E inc = 0 ,
j i ( E inc * i ) E i j = j i exp ( i k r i j ) [ ( E inc * p j ) n ̂ i j η ( r i j ) r i j ] [ ( E inc * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) ξ ( r i j ) + ( E inc * n ̂ i j ) p j γ ( r i j ) + E inc * ( n ̂ i j p j ) η ( r i j ) r i j ] ,
j i ( E i j * i ) E inc = i E inc { j i exp ( i k r i j ) [ δ * ( r i j ) ( k p j ) + η * ( r i j ) ( k n ̂ i j ) ( n ̂ i j p j ) ] } ,
k i j i ( E i k * i ) E i j = k i j i exp [ i k ( r i j r i k ) ] [ δ * ( r i k ) η ( r i j ) r i j ( p k * p j ) n ̂ i j ] + η * ( r i k ) η ( r i j ) r i j ( n ̂ i k p j ) ( n ̂ i k p k * ) n ̂ i j δ * ( r i k ) ξ ( r i j ) ( p k * n ̂ i j ) ( n ̂ i j p j ) n ̂ i j η * ( r i k ) ξ ( r i j ) ( n ̂ i k n ̂ i j ) ( n ̂ i k p k * ) ( n ̂ i j p j ) n ̂ i j + δ * ( r i k ) γ ( r i j ) ( p k * n ̂ i j ) p j + η * ( r i k ) γ ( r i j ) ( n ̂ i k n ̂ i j ) ( n ̂ i k p k * ) p j + δ * ( r i k ) η ( r i j ) r i j ( n ̂ i j p j ) p k * [ + η * ( r i k ) η ( r i j ) r i j ( n ̂ i k p k * ) ( n ̂ i j p j ) n ̂ i k ] .
δ ( r ) = k 2 r + i k r 2 1 r 3 ,
η ( r ) = k 2 r 3 i k r 2 + 3 r 3 ,
ξ ( r ) = 5 ( k 2 r 2 3 i k r 3 + 3 r 4 ) + i k 3 r k 2 r 2 ,
γ ( r ) = i k 3 r 2 k 2 r 2 3 i k r 3 + 3 r 4 .
j i exp ( i k r i j ) η ( r i j ) r i j n ̂ i j ( E inc * p j ) = λ e ̂ λ { γ E inc , γ * [ j i η ( r i j ) r i j exp ( i k r i j ) n ̂ i j , λ p j , γ ] } ,
j i exp ( i k r i j ) η ( r i j ) r i j n ̂ i j ( E inc * p j ) = λ e ̂ λ [ E inc , γ * ( j i Ξ i j λ p j , γ ) ] ,
j i exp ( i k r i j ) ξ ( r i j ) ( E inc * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) = λ e ̂ λ ( α E inc , α * { β [ j i exp ( i k r i j ) ξ ( r i j ) n ̂ i j , λ n ̂ i j , α n ̂ i j , β p j , β ] } ) ,
j i exp ( i k r i j ) ξ ( r i j ) ( E inc * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) = λ e ̂ λ { α E inc , α * [ β ( j i Ω i j λ α β p j , β ) ] } ,
j i exp ( i k r i j ) γ ( r i j ) ( E inc * n ̂ i j ) p j = λ e ̂ λ { γ E inc , γ * [ j i exp ( i k r i j ) γ ( r i j ) n ̂ i j , γ p j , λ ] } ,
j i exp ( i k r i j ) γ ( r i j ) ( E inc * n ̂ i j ) p j = λ e ̂ λ [ γ E inc , γ * ( j i Φ i j γ p j , λ ) ] ,
j i exp ( i k r i j ) η ( r i j ) r i j ( n ̂ i j p j ) E inc * = λ e ̂ λ E inc , λ * { γ [ j i exp ( i k r i j ) η ( r i j ) r i j n ̂ i j , γ p j , γ ] } ,
j i exp ( i k r i j ) η ( r i j ) r i j ( n ̂ i j p j ) E inc * = λ e ̂ λ E inc , λ * [ γ ( j i Ξ i j γ p j , γ ) ] .
Ξ i j λ = η ( r i j ) r i j exp ( i k r i j ) n ̂ i j , λ ,
Ω i j λ α β = ξ ( r i j ) exp ( i k r i j ) n ̂ i j , λ n ̂ i j , α n ̂ i j , β ,
Φ i j γ = γ ( r i j ) exp ( i k r i j ) n ̂ i j , γ .
j i ( E inc * i ) E i j = λ e ̂ λ ( { α E inc , α * [ ( j i Ξ i j λ p j , α ) β ( j i Ω i j λ α β p j , β ) + ( j i Φ i j α p j , λ ) ] } + E inc , λ * [ γ ( j i Ξ i j γ p j , γ ) ] ) .
j i ( E i j * i ) E inc = i E inc { α k α [ ( j i Γ i j p j , α ) * + β ( j i Λ i j α β p j , β ) * ] } ,
k i j i ( E i k * i ) E i j = λ e ̂ λ ( α { ( k i Γ i k p k , α ) * [ ( j i Ξ i j λ p j , α ) + ( j i Φ i j α p j , λ ) β ( j i Ω i j λ α β p j , β ) ] + [ ( j i Ξ i j λ p j , α ) + ( j i Φ i j α p j , λ ) γ ( j i Ω i j λ α γ p j , γ ) ] [ β ( k i Λ i k α β p k , β ) * ] } + [ α ( j i Ξ i j α p j , α ) ] [ ( k i Γ i k p k , λ ) * + β ( k i Λ i k λ β p k , β ) * ] ) .
Γ i j = δ ( r i j ) exp ( i k r i j ) ,
Λ i j α β = η ( r i j ) exp ( i k r i j ) n ̂ i j , α n ̂ i j , β .
E i * × B i = E inc * × B inc + j i E i j * × B inc + j i E inc * × B i j + k i j i E i k * × B i j .
B inc = k ̂ × E inc ,
B i j = k 2 exp ( i k r i j ) r i j ( 1 1 i k r i j ) ( n ̂ i j × p j )
a × ( b × c ) = b ( a c ) c ( a b ) .
E inc * × B inc = k ̂ E inc 2 ,
j i E i j * × B inc = λ e ̂ λ ( k ̂ λ { α E inc , α [ ( j i Γ i j p j , α ) * + β ( j i Λ i j α β p j , β ) * ] } E inc , λ { α k ̂ α [ ( j i Γ i j p j , α ) * + β ( j i Λ i j α β p j , β ) * ] } ) ,
j i E inc * × B i j = λ e ̂ λ { γ E inc , γ * [ ( j i Π i j λ p j , γ ) ( j i Π i j γ p j , γ ) ] } ,
k i j i E i k * × B i j = λ e ̂ λ ( α { ( k i Γ i k p k , α ) * [ ( j i Π i j λ p j , α ) ( j i Π i j α p j , λ ) ] + [ ( j i Π i j λ p j , α ) ( j i Π i j α p j , λ ) ] [ β ( k i Λ i k α β p k , β ) * ] } ) .
Π i j γ = ω ( r i j ) exp ( i k r i j ) n ̂ i j , γ ,
ω ( r i j ) = k 2 r i j ( 1 1 i k r i j ) .
( p i * i ) E i j = exp ( i k r i j ) [ ( p i * p j ) n ̂ i j η ( r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) ξ ( r i j ) + ( p i * n ̂ i j ) p j γ ( r i j ) + p i * ( n ̂ i j p j ) η ( r i j ) r i j ] ,
( E i k * i ) E i j = exp ( i k r i j ) [ ( E i k * p j ) n ̂ i j η ( r i j ) ( E i k * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) ξ ( r i j ) + ( E i k * n ̂ i j ) p j γ ( r i j ) + E i k * ( n ̂ i j p ) η ( r i j ) r i j ] .
( E i k * p j ) = exp ( i k r i k ) ( p k * p j ) δ * ( r i k ) + ( n ̂ i k p j ) ( n ̂ i k p k * ) η * ( r i k ) ,
( E i k * n i j ) = exp ( i k r i k ) ( p k * n i j ) δ * ( r i k ) + ( n ̂ i k n i j ) ( n ̂ i k p k * ) η * ( r i k ) .
E i k * × B i j = exp [ i k ( r i j r i k ) ] { [ p k * × ( n ̂ i j × p j ) ] δ * ( r i k ) ω ( r i j ) + [ n ̂ i k × ( n ̂ i j × p j ) ] ( n ̂ i k p k * ) η * ( r i k ) ω ( r i j ) } .
p k * × ( n ̂ i j × p j ) = n ̂ i j ( p k * p j ) p j ( p k * n ̂ i j ) ,
[ n ̂ i k × ( n ̂ i j × p j ) ] ( n ̂ i k × p k * ) = [ n ̂ i j ( n ̂ i k p j ) p j ( n ̂ i k n ̂ i j ) ] ( n ̂ i k p k * ) .
k i j i n ̂ i j ( p k * p j ) = λ e ̂ λ γ ( j i n ̂ i j , λ p j , γ ) ( k i p k , γ * ) ,
k i j i p j ( p k * n ̂ i j ) = λ e ̂ λ γ ( j i n ̂ i j , γ p j , λ ) ( k i p k , γ * ) ,
k i j i n ̂ i j ( n ̂ i k p j ) ( n ̂ i k p k * ) = λ e ̂ λ α ( j i n ̂ i j , λ p j , α ) [ β ( k i n ̂ i k , α n ̂ i k , β p k , β * ) ] ,
k i j i p j ( n ̂ i k n ̂ i j ) ( n ̂ i k p k * ) = λ e ̂ λ α ( j i n ̂ i j , α p j , λ ) [ β ( k i n ̂ i k , α n ̂ i k , β p k , β * ) ] .
F i , sca = 1 2 j i Re ( F i j ) ,
F i j = exp ( i k r i j ) { [ ( p i * p j ) n ̂ i j + p i * ( n ̂ i j p j ) + ( p i * n ̂ i j ) p j 5 ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) ] η ( r i j ) r i j + [ ( p i * p j ) n ̂ i j ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) ] κ ( r i j ) } .
κ ( r ) = i k 3 r k 2 r 2 ,
F i j = γ [ p i * ( j i M i j , γ p j ) e ̂ γ ] ,
M i j , γ = exp ( i k r i j ) [ ( U i j , γ + V i j , γ + W i j , γ 5 T i j , γ ) η ( r i j ) r i j + ( U i j , γ T i j , γ ) κ ( r i j ) ] ,
T i j , γ = n ̂ i j , γ ( n ̂ i j n ̂ i j ) ,
U i j , γ = n ̂ i j , γ 1 ,
V i j , γ = n ̂ i j Δ γ ,
W i j , γ = Δ γ n ̂ i j .
j i exp ( i k r i j ) ( p i * p j ) n ̂ i j η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ ( λ e ̂ λ n ̂ i j , λ ) ( γ p i , γ * p j , γ ) ] ,
j i exp ( i k r i j ) ( p i * p j ) n ̂ i j η ( r i j ) r i j = γ p i , γ * [ λ e ̂ λ ( j i Ξ i j λ p j , γ ) ] ,
j i exp ( i k r i j ) ( p i * p j ) n ̂ i j η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ λ e ̂ λ ( p i * U i j , λ p j ) ] .
j i exp ( i k r i j ) p i * ( n ̂ i j p j ) η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ ( λ e ̂ λ p i , λ * ) ( γ n ̂ i j , γ p j , γ ) ] ,
j i exp ( i k r i j ) p i * ( n ̂ i j p j ) η ( r i j ) r i j = ( λ e ̂ λ p i , λ * ) [ γ ( j i Ξ i j γ p j , γ ) ] ,
j i exp ( i k r i j ) p i * ( n ̂ i j p j ) η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ λ e ̂ λ ( p i * W i j , λ p j ) ] ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) p j η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ ( λ e ̂ λ p j , λ ) ( γ p i , γ * n ̂ i j , γ ) ] ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) p j η ( r i j ) r i j = γ p i , γ * [ λ e ̂ λ ( j i Ξ i j γ p j , λ ) ] ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) p j η ( r i j ) r i j = j i exp ( i k r i j ) η ( r i j ) r i j [ λ e ̂ λ ( p i * V i j , λ p j ) ] ,
5 j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) η ( r i j ) r i j = 5 j i exp ( i k r i j ) η ( r i j ) r i j ( λ e ̂ λ n ̂ i j , λ ) ( α p i , α * n ̂ i j , α ) ( β p j , β n ̂ i j , β ) ,
5 j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) η ( r i j ) r i j = 5 α p i , α * { λ e ̂ λ [ β ( j i Θ i j λ α β p j , β ) ] } ,
5 j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) η ( r i j ) r i j = 5 j i exp ( i k r i j ) η ( r i j ) r i j [ λ e ̂ λ ( p i * T i j , λ p j ) ] ,
Θ i j λ α β = exp ( i k r i j ) η ( r i j ) r i j n ̂ i j , λ n ̂ i j , α n ̂ i j , β ,
j i exp ( i k r i j ) ( p i * p j ) n ̂ i j κ ( r i j ) = j i exp ( i k r i j ) κ ( r i j ) [ ( λ e ̂ λ n ̂ i j , λ ) ( γ p i , γ * p j , γ ) ] ,
j i exp ( i k r i j ) ( p i * p ̂ j ) n ̂ i j κ ( r i j ) = γ p i , γ * [ λ e ̂ λ ( j i Ψ i j λ p j , γ ) ] ,
j i exp ( i k r i j ) ( p i * p ̂ j ) n ̂ i j κ ( r i j ) = j i exp ( i k r i j ) κ ( r i j ) [ λ e ̂ λ ( p i * U i j , λ p j ) ] ,
Ψ i j λ = exp ( i k r i j ) κ ( r i j ) n ̂ i j , λ ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) κ ( r i j ) = j i exp ( i k r i j ) κ ( r i j ) ( λ e ̂ λ n ̂ i j , λ ) ( α p i , α * n ̂ i j , α ) ( β n ̂ i j , β p ̂ j , β ) ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) κ ( r i j ) = α p i , α * { λ e ̂ λ [ β ( j i Σ i j λ α β p j , β ) ] } ,
j i exp ( i k r i j ) ( p i * n ̂ i j ) n ̂ i j ( n ̂ i j p j ) κ ( r i j ) = j i exp ( i k r i j ) κ ( r i j ) [ λ e ̂ λ ( p i * T i j , λ p j ) ] ,
Σ i j λ α β = exp ( i k r i j ) κ ( r i j ) n ̂ i j , λ n ̂ i j , α n ̂ i j , β .
F i , sca = 1 2 Re { γ ( p i , γ * ) [ λ e ̂ λ ( j i Ξ i j λ p j , γ ) ] + ( λ e ̂ λ p i , λ * ) [ γ ( j i Ξ i j γ p j , γ ) ] + γ ( p i , γ * ) [ λ e ̂ λ ( j i Ξ i j γ p j , λ ) ] 5 [ α ( p i , α * ) λ e ̂ λ β ( j i Θ i j λ α β p j , β ) ] + γ ( p i , γ * ) [ λ e ̂ λ ( j i Ψ i j γ p j , γ ) ] [ α ( p i , α * ) λ e ̂ λ β ( j i Σ i j λ α β p j , β ) ] } .

Metrics