Abstract

We discuss the properties of pure multipole beams with well-defined handedness or helicity, with the beam field a simultaneous eigenvector of the squared total angular momentum and its projection along the propagation axis. Under the condition of hemispherical illumination, we show that the only possible propagating multipole beams are “sectoral” multipoles. The sectoral dipole beam is shown to be equivalent to the non-singular time-reversed field of an electric and a magnetic point dipole Huygens’ source located at the beam focus. Higher order multipolar beams are vortex beams vanishing on the propagation axis. The simple analytical expressions of the electric field of sectoral multipole beams, exact solutions of Maxwell’s equations, and the peculiar behaviour of the Poynting vector and spin and orbital angular momenta in the focal volume could help to understand and model light-matter interactions under strongly focused beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Optical torque on small chiral particles in generic optical fields

Huajin Chen, Wanli Lu, Xinning Yu, Chunhua Xue, Shiyang Liu, and Zhifang Lin
Opt. Express 25(26) 32867-32878 (2017)

Analysis of Spherical Sector Resonators for the Production of a Focused Laser Beam

C. Y. She and H. Heffner
Appl. Opt. 3(6) 703-708 (1964)

Controlling the polarization singularities of the focused azimuthally polarized beams

Wei Zhang, Sheng Liu, Peng Li, Xiangyang Jiao, and Jianlin Zhao
Opt. Express 21(1) 974-983 (2013)

References

  • View by:
  • |
  • |
  • |

  1. P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University, 2015).
    [Crossref]
  2. P. Varga and P. Török, “The gaussian wave solution of maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
    [Crossref]
  3. C. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. on Microwaves, Opt. Acoust. 2, 163–166 (1978).
    [Crossref]
  4. G. Gouesbet, B. Maheu, and G. Gréhan, “Light scattering from a sphere arbitrarily located in a gaussian beam, using a bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
    [Crossref]
  5. T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
    [Crossref]
  6. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized lorenz-mie theory. i. localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
    [Crossref] [PubMed]
  7. I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011).
    [Crossref]
  8. S. M. Barnett, L. Allen, and M. J. Padgett, Optical angular momentum (CRC Press, 2016).
  9. S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
    [Crossref] [PubMed]
  10. K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
    [Crossref]
  11. K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
    [Crossref]
  12. N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
    [Crossref]
  13. X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20, 24536–24544 (2012).
    [Crossref] [PubMed]
  14. C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
    [Crossref]
  15. I. Bassett, “Limit to concentration by focusing,” Opt. Acta: Int. J. Opt. 33, 279–286 (1986).
    [Crossref]
  16. C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
    [Crossref]
  17. V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317 (1997).
    [Crossref]
  18. G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
    [Crossref] [PubMed]
  19. I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
    [Crossref]
  20. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
    [Crossref] [PubMed]
  21. T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
    [Crossref]
  22. S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
    [Crossref]
  23. I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
    [Crossref]
  24. V. V. Kotlyar and A. A. Kovalev, “Circularly polarized hankel vortices,” Opt. Express 25, 7778–7790 (2017).
    [Crossref] [PubMed]
  25. M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259 (2004).
    [Crossref]
  26. J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
    [Crossref]
  27. P. Jones, M. Rashid, M. Makita, and O. Maragò, “Sagnac interferometer method for synthesis of fractional polarization vortices,” Opt. Lett. 34, 2560–2562 (2009).
    [Crossref] [PubMed]
  28. J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
    [Crossref]
  29. F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles: theory and applications to environmental physics (Springer Science & Business Media, 2007).
  30. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).
  31. A. A. R. Neves, A. Fontes, L. A. Padilha, E. Rodriguez, C. H. de Brito Cruz, L. C. Barbosa, and C. L. Cesar, “Exact partial wave expansion of optical beams with respect to an arbitrary origin,” Opt. Lett. 31, 2477–2479 (2006).
    [Crossref] [PubMed]
  32. A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University, 1957).
    [Crossref]
  33. A. Aiello and M. Berry, “Note on the helicity decomposition of spin and orbital optical currents,” J. Opt. 17, 062001 (2015).
    [Crossref]
  34. Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
    [Crossref] [PubMed]
  35. R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
    [Crossref]
  36. J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
    [Crossref]
  37. R. Carminati, R. Pierrat, J. De Rosny, and M. Fink, “Theory of the time reversal cavity for electromagnetic fields,” Opt. Lett. 32, 3107–3109 (2007).
    [Crossref] [PubMed]
  38. S. Van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” EPL (Europhysics Lett. 25, 497 (1994).
    [Crossref]
  39. A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).
  40. G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
    [Crossref] [PubMed]
  41. J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
    [Crossref] [PubMed]
  42. D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
    [Crossref] [PubMed]
  43. M. Chen, M. Mazilu, Y. Arita, E. M. Wright, and K. Dholakia, “Dynamics of microparticles trapped in a perfect vortex beam,” Opt. Lett. 38, 4919–4922 (2013).
    [Crossref] [PubMed]
  44. T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
    [Crossref] [PubMed]
  45. R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
    [Crossref]

2019 (4)

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
[Crossref] [PubMed]

2018 (2)

T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
[Crossref] [PubMed]

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

2017 (2)

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
[Crossref] [PubMed]

V. V. Kotlyar and A. A. Kovalev, “Circularly polarized hankel vortices,” Opt. Express 25, 7778–7790 (2017).
[Crossref] [PubMed]

2015 (3)

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

A. Aiello and M. Berry, “Note on the helicity decomposition of spin and orbital optical currents,” J. Opt. 17, 062001 (2015).
[Crossref]

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

2014 (1)

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

2013 (1)

2012 (5)

X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20, 24536–24544 (2012).
[Crossref] [PubMed]

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

2011 (3)

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

2009 (1)

2008 (3)

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (3)

J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized lorenz-mie theory. i. localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. 43, 2532–2544 (2004).
[Crossref] [PubMed]

M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259 (2004).
[Crossref]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

2003 (2)

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
[Crossref]

1998 (1)

P. Varga and P. Török, “The gaussian wave solution of maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

1997 (1)

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317 (1997).
[Crossref]

1994 (2)

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

S. Van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” EPL (Europhysics Lett. 25, 497 (1994).
[Crossref]

1988 (1)

1986 (1)

I. Bassett, “Limit to concentration by focusing,” Opt. Acta: Int. J. Opt. 33, 279–286 (1986).
[Crossref]

1978 (1)

C. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. on Microwaves, Opt. Acoust. 2, 163–166 (1978).
[Crossref]

Agio, M.

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
[Crossref]

Aiello, A.

A. Aiello and M. Berry, “Note on the helicity decomposition of spin and orbital optical currents,” J. Opt. 17, 062001 (2015).
[Crossref]

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Albella, P.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Allen, L.

S. M. Barnett, L. Allen, and M. J. Padgett, Optical angular momentum (CRC Press, 2016).

Alonso, M. A.

Araneda, G.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Arita, Y.

Babiker, M.

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
[Crossref] [PubMed]

Banzer, P.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

Barbosa, L. C.

Barcikowski, S.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Barnett, S. M.

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
[Crossref] [PubMed]

S. M. Barnett, L. Allen, and M. J. Padgett, Optical angular momentum (CRC Press, 2016).

Bassett, I.

I. Bassett, “Limit to concentration by focusing,” Opt. Acta: Int. J. Opt. 33, 279–286 (1986).
[Crossref]

Bauer, T.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

Bergeret, F. S.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
[Crossref] [PubMed]

Berry, M.

A. Aiello and M. Berry, “Note on the helicity decomposition of spin and orbital optical currents,” J. Opt. 17, 062001 (2015).
[Crossref]

M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259 (2004).
[Crossref]

Blatt, R.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Bliokh, K. Y.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

K. Y. Bliokh, E. A. Ostrovskaya, M. A. Alonso, O. G. Rodríguez-Herrera, D. Lara, and C. Dainty, “Spin-to-orbital angular momentum conversion in focusing, scattering, and imaging systems,” Opt. Express 19, 26132–26149 (2011).
[Crossref]

Borghese, F.

F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles: theory and applications to environmental physics (Springer Science & Business Media, 2007).

Carminati, R.

Cesar, C. L.

Chen, M.

Chiarelli, G.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Colombe, Y.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Cortes, E.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Dainty, C.

de Brito Cruz, C. H.

De Rosny, J.

Delgado-Buscalioni, R.

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

Denti, P.

F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles: theory and applications to environmental physics (Springer Science & Business Media, 2007).

Dhayalan, V.

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317 (1997).
[Crossref]

Dholakia, K.

Dorn, R.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Edmonds, A. R.

A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University, 1957).
[Crossref]

Eismann, J. S.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

Eyraud, C.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Fernandez-Corbaton, I.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

Fink, M.

Fontes, A.

Froufe-Pérez, L.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Garcia-Camara, B.

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

García-Cámara, B.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

García-Etxarri, A.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

Gargiulo, J.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Geffrin, J.-M.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Gomez-Medina, R.

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Gómez-Medina, R.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Gonoskov, I.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

González, F.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Gouesbet, G.

Gréhan, G.

Grier, D. G.

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Heckenberg, N.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
[Crossref]

Heugel, S.

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

Higginbottom, D.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Iglesias, I.

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).

Jakobi, J.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Jones, P.

Jones, P. H.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University, 2015).
[Crossref]

König, M.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Kotlyar, V. V.

Kovalev, A. A.

Kudo, T.

T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
[Crossref] [PubMed]

Langolf, L.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Lara, D.

Larkin, K.

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

Leach, J.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Leuchs, G.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Litman, A.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Lock, J. A.

Luis-Hita, J.

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

Maheu, B.

Maier, S.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Makita, M.

Maragò, O.

Maragò, O. M.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University, 2015).
[Crossref]

Marqués, M.

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

Masuhara, H.

T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
[Crossref] [PubMed]

Mazilu, M.

Meléndez, M.

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

Mojarad, N.

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

Mojarad, N. M.

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
[Crossref]

Molina-Terriza, G.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20, 24536–24544 (2012).
[Crossref] [PubMed]

Moreno, F.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Nechayev, S.

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

Neves, A. A. R.

Nieminen, T.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
[Crossref]

Nienhuis, G.

S. Van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” EPL (Europhysics Lett. 25, 497 (1994).
[Crossref]

Nieto-Vesperinas, M.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Nori, F.

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Olmos-Trigo, J.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
[Crossref] [PubMed]

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Orlov, S.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

Ostrovskaya, E. A.

Padgett, M. J.

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
[Crossref] [PubMed]

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

S. M. Barnett, L. Allen, and M. J. Padgett, Optical angular momentum (CRC Press, 2016).

Padilha, L. A.

Peschel, U.

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

Pierrat, R.

Quabis, S.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

Rashid, M.

Rauschenbeutel, A.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Rodriguez, E.

Rodríguez-Fortuño, F.

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Rodríguez-Herrera, O. G.

Roichman, Y.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Rubinsztein-Dunlop, H.

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
[Crossref]

Ruffner, D. B.

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

Sáenz, J. J.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
[Crossref] [PubMed]

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011).
[Crossref]

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Saija, R.

F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles: theory and applications to environmental physics (Springer Science & Business Media, 2007).

Sandoghdar, V.

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
[Crossref]

Sanz-Fernández, C.

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, F. S. Bergeret, and J. J. Sáenz, “Asymmetry and spin-orbit coupling of light scattered from subwavelength particles,” Opt. Lett. 44, 1762–1765 (2019).
[Crossref] [PubMed]

Schlücker, S.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Sheppard, C.

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

C. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. on Microwaves, Opt. Acoust. 2, 163–166 (1978).
[Crossref]

Stamnes, J. J.

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317 (1997).
[Crossref]

Stefani, F. D.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Stolarski, A.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Suárez-Lacalle, I.

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

Sun, B.

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

Török, P.

P. Varga and P. Török, “The gaussian wave solution of maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

Vaillon, R.

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Van Enk, S.

S. Van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” EPL (Europhysics Lett. 25, 497 (1994).
[Crossref]

Varga, P.

P. Varga and P. Török, “The gaussian wave solution of maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

Vidal, X.

Violi, I. L.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Volpe, G.

P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University, 2015).
[Crossref]

Volz, J.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Walser, S.

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

Wright, E. M.

Yang, S.-J.

T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
[Crossref] [PubMed]

Yao, E.

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Zambrana-Puyalto, X.

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

X. Zambrana-Puyalto, X. Vidal, and G. Molina-Terriza, “Excitation of single multipolar modes with engineered cylindrically symmetric fields,” Opt. Express 20, 24536–24544 (2012).
[Crossref] [PubMed]

Zayats, A. V.

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

Zaza, C.

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

Zumofen, G.

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

Appl. Opt. (1)

EPL (Europhysics Lett. (1)

S. Van Enk and G. Nienhuis, “Spin and orbital angular momentum of photons,” EPL (Europhysics Lett. 25, 497 (1994).
[Crossref]

IEE J. on Microwaves, Opt. Acoust. (1)

C. Sheppard, “Electromagnetic field in the focal region of wide-angular annular lens and mirror systems,” IEE J. on Microwaves, Opt. Acoust. 2, 163–166 (1978).
[Crossref]

J. Mod. Opt. (1)

C. Sheppard and K. Larkin, “Optimal concentration of electromagnetic radiation,” J. Mod. Opt. 41, 1495–1505 (1994).
[Crossref]

J. Nanophotonics (1)

R. Gomez-Medina, B. Garcia-Camara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics 5, 053512 (2011).
[Crossref]

J. Opt. (1)

A. Aiello and M. Berry, “Note on the helicity decomposition of spin and orbital optical currents,” J. Opt. 17, 062001 (2015).
[Crossref]

J. Opt. A: Pure Appl. Opt. (1)

M. Berry, “Optical vortices evolving from helicoidal integer and fractional phase steps,” J. Opt. A: Pure Appl. Opt. 6, 259 (2004).
[Crossref]

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

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

T. Nieminen, H. Rubinsztein-Dunlop, and N. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79, 1005–1017 (2003).
[Crossref]

JOSA B (1)

N. M. Mojarad, V. Sandoghdar, and M. Agio, “Plasmon spectra of nanospheres under a tightly focused beam,” JOSA B 25, 651–658 (2008).
[Crossref]

Nano Lett. (1)

T. Kudo, S.-J. Yang, and H. Masuhara, “A single large assembly with dynamically fluctuating swarms of gold nanoparticles formed by trapping laser,” Nano Lett. 18, 5846–5853 (2018).
[Crossref] [PubMed]

Nat. Commun. (1)

J.-M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. Froufe-Pérez, C. Eyraud, A. Litman, R. Vaillon, F. González, M. Nieto-Vesperinas, J. J. Sáenz, and F. Moreno, “Magnetic and electric coherence in forward-and back-scattered electromagnetic waves by a single dielectric subwavelength sphere,” Nat. Commun. 3, 1171 (2012).
[Crossref]

Nat. Photonics (2)

K. Y. Bliokh, F. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9, 796 (2015).
[Crossref]

T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nat. Photonics 8, 23 (2014).
[Crossref]

Nat. Phys. (1)

G. Araneda, S. Walser, Y. Colombe, D. Higginbottom, J. Volz, R. Blatt, and A. Rauschenbeutel, “Wavelength-scale errors in optical localization due to spin–orbit coupling of light,” Nat. Phys. 15, 17 (2019).
[Crossref] [PubMed]

New J. Phys. (1)

J. Leach, E. Yao, and M. J. Padgett, “Observation of the vortex structure of a non-integer vortex beam,” New J. Phys. 6, 71 (2004).
[Crossref]

Opt. Acta: Int. J. Opt. (1)

I. Bassett, “Limit to concentration by focusing,” Opt. Acta: Int. J. Opt. 33, 279–286 (1986).
[Crossref]

Opt. Commun. (2)

I. Iglesias and J. J. Sáenz, “Scattering forces in the focal volume of high numerical aperture microscope objectives,” Opt. Commun. 284, 2430–2436 (2011).
[Crossref]

P. Varga and P. Török, “The gaussian wave solution of maxwell’s equations and the validity of scalar wave approximation,” Opt. Commun. 152, 108–118 (1998).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phil. Trans. R. Soc. A (1)

S. M. Barnett, M. Babiker, and M. J. Padgett, “Optical orbital angular momentum,” Phil. Trans. R. Soc. A 375, 20150444 (2017).
[Crossref] [PubMed]

Phys. Rev. A (3)

I. Gonoskov, A. Aiello, S. Heugel, and G. Leuchs, “Dipole pulse theory: Maximizing the field amplitude from 4 π focused laser pulses,” Phys. Rev. A 86, 053836 (2012).
[Crossref]

I. Fernandez-Corbaton, X. Zambrana-Puyalto, and G. Molina-Terriza, “Helicity and angular momentum: A symmetry-based framework for the study of light-matter interactions,” Phys. Rev. A 86, 042103 (2012).
[Crossref]

J. Olmos-Trigo, C. Sanz-Fernández, A. García-Etxarri, G. Molina-Terriza, F. S. Bergeret, and J. J. Sáenz, “Enhanced spin-orbit optical mirages from dual nanospheres,” Phys. Rev. A 99, 013852 (2019).
[Crossref]

Phys. Rev. B (1)

S. Nechayev, J. S. Eismann, G. Leuchs, and P. Banzer, “Orbital-to-spin angular momentum conversion employing local helicity,” Phys. Rev. B 99, 075155 (2019).
[Crossref]

Phys. Rev. E (1)

R. Delgado-Buscalioni, M. Meléndez, J. Luis-Hita, M. Marqués, and J. J. Sáenz, “Emergence of collective dynamics of gold nanoparticles in an optical vortex lattice,” Phys. Rev. E 98, 062614 (2018).
[Crossref]

Phys. Rev. Lett. (4)

D. B. Ruffner and D. G. Grier, “Optical forces and torques in nonuniform beams of light,” Phys. Rev. Lett. 108, 173602 (2012).
[Crossref] [PubMed]

Y. Roichman, B. Sun, A. Stolarski, and D. G. Grier, “Influence of nonconservative optical forces on the dynamics of optically trapped colloidal spheres: the fountain of probability,” Phys. Rev. Lett. 101, 128301 (2008).
[Crossref] [PubMed]

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003).
[Crossref] [PubMed]

G. Zumofen, N. Mojarad, V. Sandoghdar, and M. Agio, “Perfect reflection of light by an oscillating dipole,” Phys. Rev. Lett. 101, 180404 (2008).
[Crossref] [PubMed]

Phys. Rev. X (1)

A. Y. Bekshaev, K. Y. Bliokh, and F. Nori, “Transverse spin and momentum in two-wave interference,” Phys. Rev. X 5, 011039 (2015).

Pure Appl. Opt. (1)

V. Dhayalan and J. J. Stamnes, “Focusing of mixed-dipole waves,” Pure Appl. Opt. 6, 317 (1997).
[Crossref]

Other (6)

C. Zaza, I. L. Violi, J. Gargiulo, G. Chiarelli, L. Langolf, J. Jakobi, J. Olmos-Trigo, E. Cortes, M. König, S. Barcikowski, S. Schlücker, J. J. Sáenz, S. Maier, and F. D. Stefani, “Size-selective optical printing of silicon nanoparticles through their dipolar magnetic resonance,” ACS Photonics, doc. ID 8b01619, (posted 13 March 2019, in press).
[Crossref]

P. H. Jones, O. M. Maragò, and G. Volpe, Optical tweezers: Principles and applications (Cambridge University, 2015).
[Crossref]

S. M. Barnett, L. Allen, and M. J. Padgett, Optical angular momentum (CRC Press, 2016).

F. Borghese, P. Denti, and R. Saija, Scattering from model nonspherical particles: theory and applications to environmental physics (Springer Science & Business Media, 2007).

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1999).

A. R. Edmonds, Angular momentum in quantum mechanics (Princeton University, 1957).
[Crossref]

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 (5)

Fig. 1
Fig. 1 Poynting vector streamlines projected on the xz plane, perpendicular to the focal plane xy. The colormap illustrates the modulus of the Poynting vector, |S|, in the logarithm scale.
Fig. 2
Fig. 2 Poynting vector streamlines projected on the xz plane, perpendicular to the focal plane xy. The intensity color map corresponds to the field intensity normalised to the maximum field intensity for each case I = |E|2/|Emax|2. The maximum value is in both cases approximately given by | E max | 2 = 0.4 | E 1 , σ σ ( r = 0 ) | 2, being | E 1 , σ σ ( r = 0 ) | 2 intensity at the focal plane in the (sectoral) dipolar case.
Fig. 3
Fig. 3 Projected flow lines of the Poynting vector, P 1 , 1 + 1 and field intensities for a dipole beam. The intensity color map corresponds to the field intensity normalised to the maximum field intensity at the center I = |E|2/|Emax|2, where | E max | 2 = 8 | E 0 | 2 / 9 = | C 11 + 1 | 2 / 12 π.
Fig. 4
Fig. 4 Projected Poynting vector and colormap of the intensity for linarly polarized dipolar beams (a) on the xz plane, (b) on the yz plane and (c) on the xy focal plane. The streamlines of the Poynting vector are normal to the focal plane.
Fig. 5
Fig. 5 Projected Poynting vector, P 1 , 1 + 1, flow lines and color map of the z-component of the SAM density for a dipole beam. The toroidal structure can be easily inferred from the sign of the spin density in both planes.

Equations (41)

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

E = σ = ± 1 E σ = σ = ± 1 { l = 0 m = 1 + l C lm σ Ψ lm σ }
Ψ lm σ = 1 2 [ N lm + σ M lm ]
M lm j l ( k r ) X lm , N lm 1 k × M lm
X lm 1 l ( l + 1 ) L Y l m ( θ , φ )
Ψ lm σ = e ^ r i l ( l + 1 ) 2 j l ( k r ) kr Y l m + e ^ θ 1 2 l ( l + 1 ) { j l ( k r ) ( σ m sin θ Y l m ) + i j ˜ l ( kr ) ( Y l m θ ) } + e ^ ϕ 1 2 l ( l + 1 ) { j ˜ l ( k r ) ( m sin θ Y l m ) i σ j l ( k r ) ( Y l m θ ) } ,
j ˜ l ( kr ) j l 1 ( kr ) l j l ( kr ) kr = ( l + 1 ) j l ( kr ) kr j l + 1 ( kr ) .
S i I × , L { i r × }
S i = e ^ i S = i e ^ i × ,
J 2 Ψ lm σ = l ( l + 1 ) Ψ lm σ
J z Ψ lm σ = m Ψ lm σ
Λ Ψ lm σ 1 k × Ψ lm σ = σ Ψ lm σ
P 1 2 Re { E * × H } = σ = ± 1 σ 2 Z Im { E σ * × E σ } = 1 2 Z Re ( E + * S E + E * S E ) .
H = i Z k × ( E + + E ) = i Z ( E + E )
lim kr j j ~ sin ( k r l π / 2 ) kr ,
P lm σ ~ σ m | C lm σ | 2 4 Z ( kr ) 2 l ( l + 1 ) [ 1 sin θ | Y l m | 2 θ ] e ^ r .
P W ( l , m , σ ) = r 2 0 2 π d φ π / 2 π sin θ d θ P lm σ e ^ r = 2 π σ m | C lm σ | 2 4 Z k 2 l ( l + 1 ) | Y l m | θ = π / 2 2
e ^ r P lm σ < 0 , for π / 2 θ π . and e ^ r P lm σ > 0 , for 0 θ π / 2 .
Y l m = σ l = ( σ ) l 2 l l ! ( 2 l + 1 ) ! 4 π sin l θ e i σ l φ ,
E l , σ l σ = C l , σ l σ Ψ l , σ l σ = E 0 ( σ ) l sin l 1 θ e i σ l φ [ e ^ r i ( l + 1 ) j l kr sin θ + e ^ θ { j l + i j ˜ l cos θ } + e ^ φ { σ j ˜ l i σ j l cos θ } ]
E 0 k l Z P W π .
ξ ^ + 1 = e ^ x + i e ^ y 2 , ξ ^ 0 = e ^ z , ξ ^ 1 = e ^ x i e ^ y 2 ,
E l , σ l σ = E 0 ( σ ) l sin l 1 θ e i σ l φ 1 2 [ ξ ^ + 1 e i φ ( i j l + 1 sin 2 θ + ( σ + 1 ) [ j l cos θ i j ˜ l ] ) + ξ ^ 0 2 ( i j l + 1 cos θ + j l ) sin θ + ξ ^ 1 e i φ ( i j l + 1 sin 2 θ + ( σ 1 ) [ j l cos θ i j ˜ l ] ) ] .
E l , σ l σ ~ E 0 ( σ ) l sin l 1 θ 2 σ 2 e i σ ( l 1 ) φ [ z | z | j l i j ˜ l ] ξ ^ σ
lim kr E l , σ l σ ~ E 0 ( σ ) l sin l 1 θ 2 i σ 2 e i σ ( l 1 ) φ e i k z k | z | ξ ^ σ
E 1 , σ σ ( r = 0 ) = i 2 2 3 E 0 ξ ^ σ ,
U P W = 0 4 P W | E 1 , σ σ ( r = 0 ) | 2 = 2 3 { k 2 3 π n h c }
E 1 , σ σ = 2 i k 2 0 Im { G e e ( r ) } p ( σ ) * + 2 k 2 Z Im { G em ( r ) } m ( σ ) * ,
p ( σ ) 0 h = p ( σ ) * 0 h = 3 π k 3 E 1 , σ σ ( r = 0 ) , with m ( σ ) = m ( σ ) * = 3 π k 3 H 1 , σ σ ( r = 0 ) .
E x ( r ) = 1 2 ( E 1 , 1 1 ( r ) E 1 , 1 1 ( r ) ) and E y ( r ) = i 2 ( E 1 , 1 1 ( r ) + E 1 , 1 1 ( r ) ) .
E x , y ( r = 0 ) = i 2 2 3 E 0 e ^ x , y .
{ E l , σ l σ } * J z E l , σ l σ | E l , σ l σ | 2 = m = σ l = { E l , σ l σ } * ( L z + S z ) E l , σ l σ | E l , σ l σ | 2 = l z ( r ) + s z ( r )
{ E l , σ l σ } * S z E l , σ l σ = 2 Z σ P l , σ l σ e ^ z = μ = + 1 , 0 , 1 μ | E l , σ l σ ξ ^ μ | 2 = 2 σ | E 0 | 2 Z sin 2 l 2 θ { [ | j ˜ l | 2 + | j l | 2 ] cos 2 θ + l + 1 kr { j l j ˜ l } sin 2 θ }
P l , σ l σ = k 2 P W l π sin 2 l 2 θ [ e ^ r [ | j ˜ l | 2 + | j l | 2 ] cos θ + e ^ θ { l + 1 kr { j l j ˜ l } sin θ } + e ^ φ { ( l + 1 ) kr [ | j l | 2 σ sin θ ] } ]
E d = k 2 G e e ( r ) p ( σ ) 0 h + k 2 G e m i Z m ( σ ) = k 2 0 h ( G e e ( r ) p ( σ ) + i n h c G e m m ( σ ) )
G e e ( r ) p ( σ ) = i k 4 π { h 0 ( kr ) ( r × p ( σ ) × r r 2 + h 1 ( kr ) kr ( 3 ( r p ( σ ) ) r r 2 p ( σ ) ) } ,
G e m ( r ) m ( σ ) = i k 4 π h 1 ( k r ) r × m ( σ ) r .
E d = ( E r E θ E φ ) = i k 3 4 π 0 h ( 2 h 1 ( kr ) kr 0 0 0 h ˜ 1 ( kr ) 0 0 0 h ˜ 1 ( kr ) ) ( p r p θ p φ ) + k 3 4 π Z h 1 ( kr ) ( 0 0 0 0 0 1 0 1 0 ) ( m r m θ m φ )
h ˜ 1 ( kr ) h 0 ( kr ) h 1 ( k r ) kr .
p ( σ ) 0 h = 3 π k 3 E 1 , σ σ ( r = 0 ) = i 4 π k 3 E 0 2 ξ ^ σ = i σ 4 π k 3 E 0 2 e i σ φ ( sin θ cos θ i σ )
m ( σ ) = 3 π k 3 H 1 , σ σ ( r = 0 ) = 3 π k 3 i σ Z E 1 , σ σ ( r = 0 ) ,
E d = E 1 , σ σ = 2 i k 2 0 Im { G e e ( r ) } p ( σ ) + k 2 Z Im { G e m ( r ) } m ( σ ) = 2 i k 2 0 Im { G e e ( r ) } p ( σ ) * + k 2 Z Im { G e m ( r ) } m ( σ ) * ,

Metrics