Abstract

We present a study of the optical force on a small particle with both electric and magnetic response, immersed in an arbitrary non-dissipative medium, due to a generic incident electromagnetic field. This permits us to establish conclusions for any sign of this medium refractive index. Expressions for the gradient force, radiation pressure and curl components are obtained for the force due to both the electric and magnetic dipoles excited in the particle. In particular, for the magnetic force we tentatively introduce the concept of curl of the spin angular momentum density of the magnetic field, also expressed in terms of 3D generalizations of the Stokes parameters. From the formal analogy between the conservation of momentum and the optical theorem, we discuss the origin and significance of the electric-magnetic dipolar interaction force; this is done in connection with that of the angular distribution of scattered light and of the extinction cross section.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  6. P. C. Chaumet, and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
    [CrossRef]
  7. P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
    [CrossRef] [PubMed]
  8. M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Philos. Trans. R. Soc. Lond. A 362, 719–737 (2004).
    [CrossRef]
  9. P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
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  15. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
    [CrossRef]
  16. R. Gómez-Medina, and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. 93, 243602–243605 (2004).
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  21. B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
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  22. A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics 28, 346–353 (2008).
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  23. A. Lakhtakia, and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).
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    [CrossRef] [PubMed]
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    [CrossRef]
  36. B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
    [CrossRef]
  37. A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009).
    [CrossRef]
  38. 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]
  39. B. T. Draine, “The discrete-dipole approximation and its application to interstellar graphite grains,” Ap. J. 333, 848–872 (1988).
    [CrossRef]
  40. We acknowledge this remark in a private communication from an anonymous reviewer.
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    [CrossRef]
  45. V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
    [CrossRef]
  46. S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4, S7–S16 (2002).
    [CrossRef]
  47. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  48. . Notice that in ref. [25], the equation giving LSe has the opposite sign. However, the actual force (in S.I. units) is exactly the same as the one derived here in Gaussian units.
  49. In paraxial beams, the helicity is given by the projection of the spin on the propagation direction. If the helicity is not uniform, the curl of the spin would be, in general, different from zero. The spin curl term would then be relevant for optical beams with non-uniform helicity. However, a non-uniform spin density does not imply a non-uniform helicity: we may have non-zero curl force even if the helicity is zero, as it is the case for the standing waves with p-polarized beams discussed in [25]. (Yiqiao Tang, Harvard University, private communication).
  50. T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
    [CrossRef]

2009 (4)

P. C. Chaumet, and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009).
[CrossRef] [PubMed]

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

N. J. Moore, and M. A. Alonso, “Closed-form bases for the description of monochromatic, strongly focused, electromagnetic fields,” J. Opt. Soc. Am. A 26, 2211–2218 (2009).
[CrossRef]

A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009).
[CrossRef]

2008 (1)

A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics 28, 346–353 (2008).
[CrossRef]

2007 (3)

2006 (3)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 1339021–1339024 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
[CrossRef]

V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

2005 (1)

2004 (4)

R. Gómez-Medina, and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. 93, 243602–243605 (2004).
[CrossRef]

M. Mansuripur, “Radiation pressure and the linear momentum of the electromagnetic field,” Opt. Express 12, 5375–5401 (2004).
[CrossRef] [PubMed]

K. C. Neuman, and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

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

2003 (1)

2002 (4)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4, S7–S16 (2002).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
[CrossRef] [PubMed]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
[CrossRef]

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

2001 (1)

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, 035422–0354227 (2001).
[CrossRef]

2000 (3)

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]

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

1995 (1)

J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995).
[CrossRef]

1993 (2)

A. Lakhtakia, and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).

A. Hemmerich, and T. W. Hänsch, “Two-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410–413 (1993).
[CrossRef] [PubMed]

1992 (3)

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

1990 (1)

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

1989 (1)

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

1988 (1)

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

1986 (1)

1979 (1)

1978 (1)

1970 (1)

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

1968 (2)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of e and m,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

G. C. Sherman, “Diffracted wavefields expressible by plane wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

Albaladejo, S.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

Allen, L.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Alonso, M. A.

Alu, A.

A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009).
[CrossRef]

Arias-Gonzalez, J. R.

Ashkin, A.

Barnett, S. M.

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4, S7–S16 (2002).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Block, S. M.

K. C. Neuman, and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Burns, M. M.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Carruthers, A. E.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
[CrossRef]

Chaumet, P. C.

P. C. Chaumet, and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009).
[CrossRef] [PubMed]

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

P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
[CrossRef] [PubMed]

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, 035422–0354227 (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]

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

Chu, S.

Chylek, P.

Chýlek, P.

Cohen-Tannoudji, C.

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

Dholakia, K.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
[CrossRef]

Draine, B. T.

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

Dziedzic, J. M.

Engheta, N.

A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009).
[CrossRef]

Fournier, J. M.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Friberg, A. T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

Garcia-Camara, B.

B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
[CrossRef]

Gerz, C.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Golovchenko, J. A.

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

Gómez-Medina, R.

R. Gómez-Medina, and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. 93, 243602–243605 (2004).
[CrossRef]

Gonzalez, F.

B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
[CrossRef]

Gouesbet, G.

J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995).
[CrossRef]

Grzegorczyk, T. M.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 1339021–1339024 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
[CrossRef] [PubMed]

Hänsch, T. W.

A. Hemmerich, and T. W. Hänsch, “Two-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410–413 (1993).
[CrossRef] [PubMed]

Hemmerich, A.

A. Hemmerich, and T. W. Hänsch, “Two-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410–413 (1993).
[CrossRef] [PubMed]

Hodges, J. T.

J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995).
[CrossRef]

Jessen, P. S.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Kaivola, M.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

Kemp, B. A.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 1339021–1339024 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
[CrossRef] [PubMed]

Kiehl, J. T.

Ko, K. W.

Kong, J. A.

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 1339021–1339024 (2006).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Ab initio study of the radiation pressure on dielectric and magnetic media,” Opt. Express 13, 9280–9291 (2005).
[CrossRef] [PubMed]

Lakhtakia, A.

A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics 28, 346–353 (2008).
[CrossRef]

A. Lakhtakia, and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).

Laroche, M.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

Lett, P. D.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Lock, J. A.

J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995).
[CrossRef]

Lounis, B.

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

Mansuripur, M.

Marques, M. I.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

Moore, N. J.

Moreno, F.

B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
[CrossRef]

Mulholland, G. W.

A. Lakhtakia, and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).

Neuman, K. C.

K. C. Neuman, and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, P. C. Chaumet, and A. Rahmani, “Near field photonic forces,” Philos. Trans. R. Soc. Lond. 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]

P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
[CrossRef] [PubMed]

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, 035422–0354227 (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, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (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]

Phillips, W. D.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Pinnick, R. G.

Rahmani, A.

P. C. Chaumet, and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009).
[CrossRef] [PubMed]

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

P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
[CrossRef] [PubMed]

Ratner, M.

V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

Rolston, S. L.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Saenz, J. J.

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

Sáenz, J. J.

R. Gómez-Medina, and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. 93, 243602–243605 (2004).
[CrossRef]

Saiz, J. M.

B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
[CrossRef]

Salomon, C.

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

Setala, T.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

Sheppard, C. J. R.

Sherman, G. C.

G. C. Sherman, “Diffracted wavefields expressible by plane wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

Shevchenko, A.

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

Spreeuw, R. J. C.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Tatarkova, S. A.

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
[CrossRef]

Verkerk, P.

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of e and m,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Westbrook, C. I.

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

Woerdman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Wong, V.

V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[CrossRef]

Ap. J. (1)

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

Appl. Opt. (2)

Electromagnetics (1)

A. Lakhtakia, “Radiation pressure efficiencies of spheres made of isotropic, achiral, passive, homogeneous, negative-phase-velocity materials,” Electromagnetics 28, 346–353 (2008).
[CrossRef]

J. Electromagn. Waves Appl. (1)

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Lorentz force on dielectric and magnetic particles,” J. Electromagn. Waves Appl. 20, 827–839 (2006).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (1)

S. M. Barnett, “Optical angular-momentum flux,” J. Opt. B Quantum Semiclassical Opt. 4, S7–S16 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. A. Lock, J. T. Hodges, and G. Gouesbet, “Failure of the optical theorem for Gaussian-beam scattering by a spherical particle,” J. Opt. Soc. Am. 12, 2708–2715 (1995).
[CrossRef]

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

J. Res. Natl. Inst. Stand. Technol. (1)

A. Lakhtakia, and G. W. Mulholland, “On two numerical techniques for light scattering by dielectric agglomerated structures,” J. Res. Natl. Inst. Stand. Technol. 98, 699–716 (1993).

Opt. Express (4)

Opt. Lett. (2)

Philos. Trans. R. Soc. Lond. A (1)

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

Phys. Rev. A (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (4)

V. Wong, and M. Ratner, “Gradient and nongradient contributions to plasmon-enhanced optical forces on silver nanoparticles,” Phys. Rev. B 73, 0754161–0754166 (2006).
[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]

P. C. Chaumet, and M. Nieto-Vesperinas, “Electromagnetic force on a metallic particle in the presence of a dielectric surface,” Phys. Rev. B 62, 11185–11191 (2000).
[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, 035422–0354227 (2001).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Setala, A. Shevchenko, M. Kaivola, and A. T. Friberg, “Degree of polarization for optical near fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66, 0166151–0166158 (2002).
[CrossRef]

Phys. Rev. Lett. (13)

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, “One-Dimensional Optically Bound Arrays of Microscopic Particles,” Phys. Rev. Lett. 89, 283901–283904 (2002).
[CrossRef]

R. Gómez-Medina, and J. J. Sáenz, “Unusually Strong Optical Interactions between Particles in Quasi-One-Dimensional Geometries,” Phys. Rev. Lett. 93, 243602–243605 (2004).
[CrossRef]

A. Ashkin, “Acceleration and Trapping of Particles by Radiation Pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
[CrossRef]

B. A. Kemp, T. M. Grzegorczyk, and J. A. Kong, “Optical momentum transfer to absorbing Mie particles,” Phys. Rev. Lett. 97, 1339021–1339024 (2006).
[CrossRef]

P. C. Chaumet, A. Rahmani, and M. Nieto-vesperinas, “Optical trapping and manipulation of nano-objects with an apertureless probe,” Phys. Rev. Lett. 88, 123601–123604 (2002).
[CrossRef] [PubMed]

P. Verkerk, B. Lounis, C. Salomon, and C. Cohen-Tannoudji, “Dynamics and spatial order of cold cesium atoms in a periodic optical potential,” Phys. Rev. Lett. 68, 3861–3864 (1992).
[CrossRef]

P. S. Jessen, C. Gerz, P. D. Lett, W. D. Phillips, S. L. Rolston, R. J. C. Spreeuw, and C. I. Westbrook, “Observation of quantized motion of Rb atoms in an optical field,” Phys. Rev. Lett. 69, 49–52 (1992).
[CrossRef] [PubMed]

A. Hemmerich, and T. W. Hänsch, “Two-dimensional atomic crystal bound by light,” Phys. Rev. Lett. 70, 410–413 (1993).
[CrossRef] [PubMed]

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Binding,” Phys. Rev. Lett. 63, 1233–1236 (1989).
[CrossRef] [PubMed]

S. Albaladejo, M. I. Marques, M. Laroche, and J. J. Saenz, “Scattering Forces from the Curl of the Spin Angular momentum of a Light Field,” Phys. Rev. Lett. 102, 1136021–1136024 (2009).
[CrossRef]

G. C. Sherman, “Diffracted wavefields expressible by plane wave expansions containing only homogeneous waves,” Phys. Rev. Lett. 21, 761–764 (1968).
[CrossRef]

B. Garcia-Camara, F. Moreno, F. Gonzalez, and J. M. Saiz, “Comment on Experimental Evidence of Zero Forward Scattering by Magnetic Spheres,” Phys. Rev. Lett. 98, 179701–179704 (2007).
[CrossRef]

A. Alu, and N. Engheta, “Cloaking a Sensor,” Phys. Rev. Lett. 102, 233901–233904 (2009).
[CrossRef]

Rev. Sci. Instrum. (1)

K. C. Neuman, and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004).
[CrossRef]

Science (1)

M. M. Burns, J. M. Fournier, and J. A. Golovchenko, “Optical Matter: Crystallization and Binding in Intense Optical Fields,” Science 249, 749–754 (1990).
[CrossRef] [PubMed]

Sov. Phys. Usp. (1)

V. G. Veselago, “The electrodynamics of substances with simultaneous negative values of e and m,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Other (10)

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media, 2nd revised edition (Pergamon Press, Oxford, 1984).

. Notice that in ref. [25], the equation giving LSe has the opposite sign. However, the actual force (in S.I. units) is exactly the same as the one derived here in Gaussian units.

In paraxial beams, the helicity is given by the projection of the spin on the propagation direction. If the helicity is not uniform, the curl of the spin would be, in general, different from zero. The spin curl term would then be relevant for optical beams with non-uniform helicity. However, a non-uniform spin density does not imply a non-uniform helicity: we may have non-zero curl force even if the helicity is zero, as it is the case for the standing waves with p-polarized beams discussed in [25]. (Yiqiao Tang, Harvard University, private communication).

L. Novotny, and B. Hecht, Principles of Nano-Optics, (Cambridge University Press, Cambridge, 2006).

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

M. Nieto-Vesperinas, Scattering and Diffraction in Physical Optics, (2nd edition, World Scientific, Singapore, 2006).

We acknowledge this remark in a private communication from an anonymous reviewer.

C. F. Bohren, and D. R. Huffman, Absorption and Scattering of Light by Small Particles, (John Wiley, New York, 1998).
[CrossRef]

M. Born, and E. Wolf, Principles of Optics, 7 th edition, Cambridge U.P., Cambridge, 1999.

J. D. Jackson, Classical Electrodynamics, (3rd edition, John Wiley, New York, 1998).

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Figures (4)

Fig. 1.
Fig. 1.

Scattering by a dipolar particle. Left: Scattering geometry. Top right: Angular θ-distribution of scattered light, (integrated over the azimuthal angle ϕ), by an electric dipolar particle. Bottom right: The same for a small perfectly conducting spherical particle (which has both electric and magnetic induced dipoles). The asymmetry of the radiation from this particle leads to the interaction force term < F e-m > (see text).

Fig. 2.
Fig. 2.

Normalized real and imaginary parts of both electric and magnetic static polarizabilities, Eq. (36) and Eq. (37), in terms of the wavelength, for a 100nm highly conducting spherical particle in a host medium with ε = μ = 1. The peaks of both real and imaginary parts of the magnetic polarizability are about 30 a.u

Fig. 3.
Fig. 3.

Normalized real and imaginary parts of both electric and magnetic radiation-reaction polarizabilities, Eq. (34) and Eq. (35) in terms of the wavelength, for a 100nm highly conducting spherical particle in a host medium with ε = μ = 1. As λ becomes very large, the real part of the electric and magnetic polarizability tends to the perfect conductor values: 1and-1/2 a.u

Fig. 4.
Fig. 4.

The three components of the total radiation pressure versus the wavelength for the same particle as in Figs. 2 and 3, in vacuum. Forces are normalized to the incident field intensity: ∣e (i)2. Notice that as λ grows, these components tend to zero, [cf. Eq. (41)].

Equations (64)

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

< F > = 1 8 π { s [ ε ( E · s ) E * + μ 1 ( B · s ) B * 1 2 ( ε E 2 + μ 1 B 2 ) s ] dS } ,
< F > = 1 2 { p ( E ( i ) * ) + m ( B ( i ) * ) 2 k 4 3 μ ε ( p × m * ) } .
B ( r ) ( r ) = k 2 ( μ ( s × m ) × s + μ ε s × p ) e ikr r , E ( r ) ( r ) = 1 n B ( r ) ( r ) × s
< F > = n c S { S S ( i ) } dS .
S S ( i ) = c 8 π μ 1 { E ( r ) × B ( i ) + E ( i ) × B ( i ) * + E ( r ) × B ( r ) * }
= c 8 πn { ε ( E ( i ) · E ( r ) * ) + μ 1 ( B ( r ) · B ( i ) ) 1 2 ( ε E ( r ) 2 + μ 1 B ( r ) 2 ) } s
c 8 πn { ε ( E ( i ) · s ) E ( r ) * + μ 1 ( B ( i ) * · s ) B ( r ) } .
E ( i ) ( r ) = 𝓓 e ( i ) ( u ) exp ( ik u · r ) d Ω ,
B ( i ) ( r ) = 𝓓 b ( i ) ( u ) exp ( ik u · r ) d Ω ,
· B ( i ) ( r ) = 0 ,
· E ( i ) ( r ) = ik 𝓓 ( e ( i ) ( u ) · u ) exp ( ik u · r ) d Ω = 0 .
< F > = 1 8 π s { ε E ( i ) * · E ( r ) + μ 1 B ( i ) * · B ( r ) + 1 2 ( ε E ( r ) 2 + μ 1 B ( r ) 2 ) } s dS .
< F e > + < F m > = k 8 π { 𝓓 i u ( ε e ( i ) * ( u ) · p + μ 1 b ( i ) ( u ) · m ) d Ω }
= 1 2 { p ( E ( i ) * ) r = r 0 + m ( B ( i ) * ) r = r 0 } .
< F e m > = n c s S ( r ) dS = k 4 3 μ ε { p × m * } ,
( a ) = s { S S ( i ) } · s dS
= c 8 π { s ε E ( i ) * · E ( r ) + μ 1 B ( i ) * · B ( r ) + 1 2 ( ε E ( r ) 2 + μ 1 B ( r ) 2 ) dS } .
( a ) = ω 2 { p · E ( i ) * ( r 0 ) } ω 2 { m · B ( i ) * ( r 0 ) } + c n k 4 3 { ε 1 p 2 + μ m 2 } .
( a ) + ( s ) = ω 2 { p · E ( i ) * ( r 0 ) } + ω 2 { m · B ( i ) * ( r 0 ) } .
S ( r ) dS = c 8 πn k 4 ( ε 1 p × s 2 + μ m × s 2 ) s d Ω
+ c 4 πn k 4 μ ε { ( s × p ) · m * } s d Ω ,
p = α e e ( i ) ; m = α m b ( i ) ,
σ ( ext ) = 4 πk [ ε 1 α e ] + 4 πk [ μ α m ] = σ ( a ) + σ ( s ) .
σ ( s ) = ( n c 8 π ε e ( i ) 2 ) s S ( r ) · s dS = s ( d σ ( s ) d Ω ) d Ω ,
d σ ( s ) d Ω = k 4 ε 1 α e 2 ( sin 2 ϕ + cos 2 θ cos 2 ϕ )
+ k 4 μα m 2 ( cos 2 ϕ + cos 2 θ sin 2 ϕ )
+ 2 k 4 μ ε 2 { α e α m * } cos θ .
σ ( s ) = 8 π 3 k 4 ( ε 1 α e 2 + μα m 2 ) ,
< F > = < F e > + < F m > + < F e m >
= k 2 { p · e ( i ) * + m · b ( i ) * } n c s S ( r ) dS
= ε 8 π e ( i ) 2 { σ ( ext ) k k s ( d σ ( s ) ) s d Ω } ,
< F > = k 2 ε e ( i ) 2 { α e ε + μℑ α m 2 k 3 3 μ ε ( α e α m + α e α m ) } .
α e = α e ( 0 ) ( 1 i 2 3 ε k 3 α e ( 0 ) ) 1 ,
α e ( 0 ) = ε a 3 ε p ε ε p + 2 ε .
σ e ( ext ) = 4 πkℑ [ ε 1 α e ] = 8 π k 4 3 ε 2 α e 2 8 π k 4 3 ε 2 α e ( 0 ) 2 .
α m = α m ( 0 ) ( 1 i 2 3 μ k 3 α m ( 0 ) ) 1 ,
α m ( 0 ) = μ 1 a 3 μ p μ μ p + 2 μ .
σ m ( ext ) = 4 πkℑ [ μ α m ] = 8 π k 4 μ 2 3 α m 2 8 π k 4 μ 2 3 α m ( 0 ) 2 .
< F > k 4 a 6 3 k k ε e ( i ) 2 { [ ε p ε ε p + 2 ε ] 2 + [ μ p μ μ p + 2 μ ] 2 ε p ε ε p + 2 ε μ p μ μ p + 2 μ } .
α e = i 3 ε 2 k 3 a 1 ,
α m = i 3 2 μ k 3 b 1 .
α e ( 0 ) = 3 ε 2 k 3 μ n p 2 j 1 ( n p x ) [ x j 1 ( x ) ] μ p j 1 ( x ) [ n p x j 1 ( n p x ) ] μ n p 2 j 1 ( n p x ) [ x y 1 ( x ) ] μ p y 1 ( x ) [ n p x j 1 ( n p x ) ] ,
α m ( 0 ) = 3 2 μ k 3 μ p j 1 ( n p x ) [ x j 1 ( x ) ] μ j 1 ( x ) [ n p x j 1 ( n p x ) ] μ p j 1 ( n p x ) [ x y 1 ( x ) ] μ y 1 ( x ) [ n p x j 1 ( n p x ) ] .
ε p ( ω ) = 1 + i 4 πσ ω
d σ ( s ) d Ω = k 4 a 6 ( 5 4 ( cos 2 ϕ + 1 4 sin 2 ϕ ) sin 2 θ cos θ )
d σ ( s ) d Ω = 5 8 k 4 a 6 ( 1 + cos 2 θ 8 5 cos θ ) .
< F > = k 4 a 6 3 k k ε e ( i ) 2 { 1 + 1 4 + 1 2 } = k k 7 k 4 a 6 12 ε e ( i ) 2 .
< F e > = 1 4 α e E 2 + k 2 n α e ( E × B * ) + 1 2 α e [ ( E * · ) E ] .
< F m > = 1 4 α m B 2 + k 2 n α m ( E × B * ) + 1 2 α m [ ( B * · ) B ] .
< F e m > = k 4 3 μ ε { ( α e α m * ) ( E × B * ) ( α e α m * ) ( E × B * ) }
= k 4 3 μ ε ( α e α m * ) ( E × B * ) + k 3 . 3 μ ( α e α m * ) [ 1 2 E 2 [ ( E * · ) E ] ] .
F e = 4 πℜ { α e } ε U e + σ e ext { n c S } σ e ext { c n × L Se }
U e = 1 2 ε 8 π E 2 .
S = 1 2 c 4 πμ E × B * .
L Se = 1 2 ε 8 πωi E * × E
1 2 α m [ ( B * · ) B ] = i α m 4 × ( B × B * ) ,
F m = 4 πℜ { α m } ε U m + α m ext { n c S } σ m ext { c n × L Sm }
U m = 1 2 1 8 πμ B 2 ;
L Sm = 1 2 1 8 πμωi B * × B
s 2 = ( 3 / 2 ) i ( Φ xy Φ yx ) ,
s 5 = ( 3 / 2 ) i ( Φ xz Φ zx ) ,
s 7 = ( 3 / 2 ) i ( Φ yz Φ zy ) ,
1 2 = α e [ ( E * · ) E ] = 1 6 α e × ( s 7 e s 5 e , s 2 e ) ,
1 2 = α m [ ( B * · ) B ] = 1 6 α m × ( s 7 e s 5 e , s 2 e ) .

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