Abstract

Plasmonic nanoparticles are commonly used to tune and direct the radiation from electric dipolar emitters. Less progress has been made towards understanding complementary systems of magnetic nature. However, it has been recently shown that high-index dielectric spheres can act as effective magnetic antennas. Here we explore the concept of coupling dielectric magnetic antennas with either an electric or magnetic dipolar emitter in a similar fashion to the purely electric systems reported previously. We investigate the enhancement of radiation from systems comprising admixtures of these electric and magnetic elements and perform a full study of its dependence on the distance and polarization of the emitter with respect to the antenna. A comparison to the plasmon antennas reveals remarkable symmetries between electric and magnetic systems, which might lead to novel paradigms in the design of nanophotonic devices that involve magnetic activity.

© 2012 OSA

Full Article  |  PDF Article

Errata

M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suárez-Lacalle, S. Mackowski, and J. Aizpurua, "Dielectric antennas - a suitable platform for controlling magnetic dipolar emission: errata," Opt. Express 20, 18609-18610 (2012)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-20-17-18609

References

  • View by:
  • |
  • |
  • |

  1. E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).
  2. M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
    [CrossRef] [PubMed]
  3. A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
    [CrossRef]
  4. S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
    [CrossRef] [PubMed]
  5. P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
    [CrossRef] [PubMed]
  6. R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
    [CrossRef] [PubMed]
  7. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
    [CrossRef]
  8. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
    [CrossRef] [PubMed]
  9. R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
    [CrossRef]
  10. Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
    [CrossRef]
  11. R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
    [CrossRef]
  12. H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
    [CrossRef]
  13. G. Colas des Francs, A. Bouhelier, E. Finot, J. C. Weeber, A. Dereux, C. Girard, and E. Dujardin, “Fluorescence relaxation in the nearfield of a mesoscopic metallic particle: distance dependence and role of plasmon modes,” Opt. Express 16, 17654–17666 (2008).
    [CrossRef] [PubMed]
  14. R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
    [CrossRef] [PubMed]
  15. J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
    [CrossRef] [PubMed]
  16. R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
    [CrossRef]
  17. S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett. 106, 193004 (2011).
    [CrossRef] [PubMed]
  18. A. Alú and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009).
    [CrossRef] [PubMed]
  19. N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
    [CrossRef] [PubMed]
  20. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
    [CrossRef] [PubMed]
  21. R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
    [CrossRef]
  22. V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15, 61–73 (2005).
  23. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, 1998).
    [CrossRef]
  24. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).
  25. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).
  26. In the absence of absorption, ℜ(an) = |an|2 and ℜ(bn) = |bn|2, where ℜ(z) denotes the real part of z.
  27. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [CrossRef]
  28. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

2011 (5)

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett. 106, 193004 (2011).
[CrossRef] [PubMed]

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
[CrossRef] [PubMed]

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

2010 (2)

R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

2009 (2)

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

A. Alú and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009).
[CrossRef] [PubMed]

2008 (3)

G. Colas des Francs, A. Bouhelier, E. Finot, J. C. Weeber, A. Dereux, C. Girard, and E. Dujardin, “Fluorescence relaxation in the nearfield of a mesoscopic metallic particle: distance dependence and role of plasmon modes,” Opt. Express 16, 17654–17666 (2008).
[CrossRef] [PubMed]

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

2007 (1)

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

2006 (3)

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

2005 (1)

V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15, 61–73 (2005).

1992 (1)

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

1991 (1)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

1988 (1)

Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
[CrossRef]

1982 (1)

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
[CrossRef]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

Aizpurua, J.

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
[CrossRef] [PubMed]

Alú, A.

Anger, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Artuso, R. D.

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

Bao, K.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Bharadwaj, P.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Bohren, C. F.

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

Bouhelier, A.

Bowden, C. M.

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Brown, L. V.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Bryant, G. W.

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

Carminati, R.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Chantada, L.

Chizhik, A.

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Colas des Francs, G.

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

Dereux, A.

Dorfmüller, J.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Dowling, J. P.

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Dujardin, E.

Engheta, N.

Esteban, R.

R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Feldmann, J.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Finot, E.

Froufe-Pérez, L. S.

García-Cámara, B.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

Garcia-Etxarri, A.

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

García-Etxarri, A.

George, T. F.

Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
[CrossRef]

Girard, C.

Glauber, R. J.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Gómez-Medina, R.

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
[CrossRef] [PubMed]

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

González, F.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

Greffet, J.

R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Greffet, J.-J.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Gutbrod, R.

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

Hakanson, U.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Halas, N. J.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Henkel, C.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Huffman, D. R.

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

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Karaveli, S.

S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett. 106, 193004 (2011).
[CrossRef] [PubMed]

Khoptyar, D.

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

Kim, Y. S.

Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
[CrossRef]

Klar, T. A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Klimov, V. V.

V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15, 61–73 (2005).

Koenderink, A. F.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Kreuzer, M. P.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

Kühn, S.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Kürzinger, K.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Letokhov, V. S.

V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15, 61–73 (2005).

Leung, P. T.

Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
[CrossRef]

Lewenstein, M.

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

Liu, N.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

López, C.

Meixner, A. J.

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

Mertens, H.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Mishchenko, M. I.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

Moreno, F.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

Mukherjee, S.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Nichtl, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
[CrossRef] [PubMed]

Nordlander, P.

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Novotny, L.

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

Polman, A.

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

Purcell, E. M.

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

Ringler, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Rogobete, L.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Ruppin, R.

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
[CrossRef]

Sáenz, J. J.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron Silicon particles in the infrared,” Opt. Express 19, 4815–4862 (2011).
[CrossRef] [PubMed]

Sandoghdar, V.

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

Scheffold, F.

Schleifenbaum, F.

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

Schwemer, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Suárez-Lacalle, I.

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

Taminiau, T. H.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Teperik, T.

R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Travis, L. D.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

van Hulst, N. F.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Vigoureux, J. M.

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Volpe, G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

Weeber, J. C.

Wunderlich, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Zia, R.

S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett. 106, 193004 (2011).
[CrossRef] [PubMed]

J. Chem. Phys. (1)

R. Ruppin, “Decay of an excited molecule near a small metal sphere,” J. Chem. Phys. 76, 1681–1684 (1982).
[CrossRef]

J. Nanophoton. (1)

R. Gómez-Medina, B. García-Cámara, 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. Nanophoton. 5, 053512 (2011).
[CrossRef]

Laser Phys. (1)

V. V. Klimov and V. S. Letokhov, “Electric and magnetic dipole transitions of an atom in the presence of spherical dielectric interface,” Laser Phys. 15, 61–73 (2005).

Nano Lett. (1)

N. Liu, S. Mukherjee, K. Bao, L. V. Brown, J. Dorfmüller, P. Nordlander, and N. J. Halas, “Magnetic plasmon formation and propagation in artificial aromatic molecules,” Nano Lett. 12, 364–369 (2011).
[CrossRef] [PubMed]

Nat. Photonics (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Van Hulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2, 234–237 (2008).
[CrossRef]

Opt. Commun. (1)

R. Carminati, J.-J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun. 261, 368–375 (2006).
[CrossRef]

Opt. Express (3)

Phys. Rev. (1)

E. M. Purcell, “Spontaneous emission probabilities at radio frequencies,” Phys. Rev. 69, 681–681 (1946).

Phys. Rev. A (2)

R. J. Glauber and M. Lewenstein, “Quantum optics of dielectric media,” Phys. Rev. A 43, 467–491 (1991).
[CrossRef] [PubMed]

J. P. Dowling and C. M. Bowden, “Atomic emission rates in inhomogeneous media with applications to photonic band structures,” Phys. Rev. A 46, 612–622 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (3)

R. D. Artuso, G. W. Bryant, A. Garcia-Etxarri, and J. Aizpurua, “Using local fields to tailor hybrid quantum dot-metal nanoparticle systems: connecting the dots,” Phys. Rev. B 83, 235406 (2011).
[CrossRef]

H. Mertens, A. F. Koenderink, and A. Polman, “Plasmon-enhanced luminescence near noble-metal nanospheres: comparison of exact theory and an improved Gersten and Nitzan model,” Phys. Rev. B 76, 115123 (2007).
[CrossRef]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[CrossRef]

Phys. Rev. Lett. (6)

S. Karaveli and R. Zia, “Spectral tuning by selective enhancement of electric and magnetic dipole emission,” Phys. Rev. Lett. 106, 193004 (2011).
[CrossRef] [PubMed]

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

A. Chizhik, F. Schleifenbaum, R. Gutbrod, A. Chizhik, D. Khoptyar, and A. J. Meixner, “Tuning the fluorescence emission spectra of a single molecule with a variable optical subwavelength metal microcavity,” Phys. Rev. Lett. 201, 073002 (2009).
[CrossRef]

S. Kühn, U. Hakanson, L. Rogobete, and V. Sandoghdar, “Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna,” Phys. Rev. Lett. 97, 017402 (2006).
[CrossRef] [PubMed]

P. Anger, P. Bharadwaj, and L. Novotny, “Enhancement and quenching of single-molecule fluorescence,” Phys. Rev. Lett. 96, 113002 (2006).
[CrossRef] [PubMed]

R. Esteban, T. Teperik, and J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett. 104, 026802 (2010).
[CrossRef] [PubMed]

Science (1)

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010).
[CrossRef] [PubMed]

Surf. Sci. (1)

Y. S. Kim, P. T. Leung, and T. F. George, “Classical decay rates for molecules in the presence of a spherical surface: a complete treatment,” Surf. Sci. 195, 1–14 (1988).
[CrossRef]

Other (5)

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999).

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

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge Univ. Press, 2002).

In the absence of absorption, ℜ(an) = |an|2 and ℜ(bn) = |bn|2, where ℜ(z) denotes the real part of z.

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

Spectral features of the scattering cross section of a 230 nm radius Si nanosphere in vacuum illuminated by a plane wave (a) and the decay rate enhancements of an electric or magnetic emitter positioned in the vicinity of the particle (c). The schematics of the systems are shown in (b) and (d), respectively. In (a), the total scattering efficiency Cs is separated into different contributions: electric (a1) and magnetic dipolar (b1), electric (a2) and magnetic quadrupolar (b2), according to Eq. (11). Spectra of the radiative rate enhancements shown in (c) were calculated using Eqs. (2),(4),(6),(8) for the electric (solid lines) and magnetic (dashed lines) emitter. The emitter is oriented either perpendicularly (blue lines) or parallelly (red lines) with respect to the closest surface of the sphere. The refractive index of the silicon nanosphere is 3.5, while the distance from the emitter to the surface of the sphere is set to 50 nm.

Fig. 2
Fig. 2

Spectra of the decay rate enhancements of electric (pemi, (a,b)) or magnetic (memi, (c,d)) nature near the silicon sphere of varying radius a in vacuum. Dipoles are oriented either perpendicularly (a,c) or parallelly (b,d) to the surface of the antenna, positioned at the fixed distance of 50 nm from its surface. Dashed lines correspond to the Mie resonances an and bn, as denoted in each of the plots. Geometries of the setup are shown in the schematics.

Fig. 3
Fig. 3

Distribution of the electric (upper row) and magnetic (lower row) induced field amplitudes by an electric (pemi) and magnetic (memi) dipole at the 50 nm distance from the surface of a 230 nm radius silicon sphere. Shown cross-sections contain the dipole and the center of the sphere. Plots (a–d) correspond to the dipolar electric mode at λ = 1350 nm, with the induced dipole denoted as pind, excited by the perpendicular electric (a,b) or parallel magnetic (c,d) emitter. (e–h) illustrate the field distributions at the dipolar magnetic mode λ = 1680 nm (mind), induced by the parallel electric (e,f) or perpendicular magnetic (g,h) emitter. Schematics of the exciting and induced dipoles are shown at the top of the figure. Intensities of the induced fields Eind and Hind are normalized to the values of the fields E0 and H0 produced by the dipolar emitter in the absence of the particle and evaluated at the distance of 280 nm from the emitter in the direction perpendicular to its axis. The position of the normalization point is marked by a white cross in each case.

Fig. 4
Fig. 4

Spectral features of the scattering efficiency of a 50 nm radius silver nanosphere in vacuum illuminated by a plane wave (a) and the radiative (c) and total (d) decay rate enhancements of an electric and magnetic emitter positioned in the vicinity of the particle. The geometries of the systems for calculating the plane wave extinction and the properties of dipolar emission are depicted in (b) and (e), respectively. In (a), the total scattering efficiency Cs is split into different contributions: electric dipolar (a1), magnetic dipolar (b1), electric quadrupolar (b2), according to Eq. (11). In plots (c) and (d), the enhancement factors of the, respectively, radiative and total decay rates of the dipoles are shown. In both plots, the spectra were calculated for the electric (solid lines) and magnetic (dashed lines) emitters. The orientation of the emitter with respect to the surface of the sphere is either perpendicular (blue lines) or parallel (red lines). The refractive index of the uniform silver nanosphere is taken from [27].

Fig. 5
Fig. 5

Distance dependence of the decay rate enhancements calculated using the exact formulas (solid lines) and the dipolar interaction approach (dashed lines). In plots (a–d), the emitter is positioned near the 230 nm radius silicon sphere and for (e–f) - near the 50 nm radius silver sphere. The wavelength of radiation matches the dipolar electric mode at 1350 nm for silicon (a–b) and 420 nm for silver sphere (e–f) or the dipolar magnetic mode in silicon antenna at 1680 nm (c–d). In each case, only the dominant induced dipole is considered for the dipolar approximation, while the mode of the complementary nature (magnetic or electric) is neglected. The insets show the orientation and the electric of magnetic nature of both the emitter (pemi or memi, respectively) and the induced dipolar mode (pind or mind, respectively) in the antenna. The distance is measured between the dipole and the center of the antenna.

Equations (40)

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

Γ RAD , e Γ 0 = 3 2 n = 0 ( 2 n + 1 ) n ( n + 1 ) | j n ( k z ) a n h n ( 1 ) ( k z ) k z | 2
Γ TOT , e Γ 0 = 1 3 2 n = 0 ( 2 n + 1 ) n ( n + 1 ) a n [ h n ( 1 ) ( k z ) k z ] 2
Γ RAD || , e Γ 0 = 3 4 n = 0 ( 2 n + 1 ) [ | j n ( k z ) b n h n ( 1 ) ( k z ) | 2 + | ψ n ' ( k z ) a n ζ n ' ( k z ) k z | 2 ] 2
Γ TOT || , e Γ 0 = 1 3 4 n = 0 ( 2 n + 1 ) [ a n [ ζ n ' ( k z ) k z ] 2 + b n [ h n ( 1 ) ( k z ) ] 2 ]
Γ RAD , m Γ 0 = 3 2 n = 0 ( 2 n + 1 ) n ( n + 1 ) | j n ( k z ) b n h n ( 1 ) ( k z ) k z | 2
Γ TOT , m Γ 0 = 1 3 2 n = 0 ( 2 n + 1 ) n ( n + 1 ) b n [ h n ( 1 ) ( k z ) k z ] 2
Γ RAD || , m Γ 0 = 3 4 n = 0 ( 2 n + 1 ) [ | j n ( k z ) a n h n ( 1 ) ( k z ) | 2 + | ψ n ' ( k z ) b n ζ n ' ( k z ) k z | 2 ] 2
Γ TOT || , m Γ 0 = 1 3 4 n = 0 ( 2 n + 1 ) [ b n [ ζ n ' ( k z ) k z ] 2 + a n [ h n ( 1 ) ( k z ) ] 2 ] .
a n = M ψ n ( k 1 a ) ψ n ' ( k a ) ψ n ( k a ) ψ n ' ( k 1 a ) M ψ n ( k 1 a ) ζ n ' ( k a ) ζ n ( k a ) ψ n ' ( k 1 a )
b n = ψ n ( k 1 a ) ψ n ' ( k a ) M ψ n ( k a ) ψ n ' ( k 1 a ) ψ n ( k 1 a ) ζ n ' ( k a ) M ζ n ( k a ) ψ n ' ( k 1 a )
σ s = 2 π k 2 n = 1 ( 2 n + 1 ) ( ( a n ) + ( b n ) ) = n = 1 ( σ s ( a n ) + σ s ( b n ) )
Γ TOT , e Γ 0 | dip = 1 + 3 k 3 2 π [ α E e 2 i k z ( 1 2 i k z + 1 ( k z ) 2 ) ]
Γ TOT | | , e Γ 0 | dip = 1 + 3 k 3 8 π [ α E e 2 i k z 1 ( k z ) 2 ( 1 + 2 i k z 3 ( k z ) 2 2 i ( k z ) 3 + 1 ( k z ) 4 ) ] + 3 k 3 8 π [ α M e 2 i k z 1 ( k z ) 2 ( 1 2 i k z + 1 ( k z ) 2 ) ]
Γ TOT , m Γ 0 | dip = 1 + 3 k 3 2 π [ α M e 2 i k z ( 1 2 i k z + 1 ( k z ) 2 ) ]
Γ TOT | | , m Γ 0 | dip = 1 + 3 k 3 8 π [ α M e 2 i k z 1 ( k z ) 2 ( 1 + 2 i k z 3 ( k z ) 2 2 i ( k z ) 3 + 1 ( k z ) 4 ) ] + 3 k 3 8 π [ α E e 2 i k z 1 ( k z ) 2 ( 1 2 i k z + 1 ( k z ) 2 ) ] ,
H dip ( r , ω ) = { ν D ν [ p ν M ν ( 1 ) ( k r ) + q ν N ν ( 1 ) ( k r ) ] , r > r ν D ν [ s ν M ν ( 3 ) ( k r ) + t ν N ν ( 3 ) ( k r ) ] , r < r
ν = σ = o , e n = 1 m = 0 n ,
D ν = δ m ( 2 n + 1 ) ( n m ) ! 4 n ( n + 1 ) ( n + m ) !
δ m = { 1 , m = 0 2 , m > 0 .
s ν = i k 3 π M ν ( 1 ) ( k r ) m
t ν = i k 3 π N ν ( 1 ) ( k r ) m
p ν = i k 3 π M ν ( 3 ) ( k r ) m
q ν = i k 3 π N ν ( 3 ) ( k r ) m .
E = Z H p , H = 1 Z E p
H sc ( r , ω ) = ν D ν [ u ν M ν ( 3 ) ( k r ) + v ν N ν ( 3 ) ( k r ) ]
H tr ( r , ω ) = ν D ν [ f ν M ν ( 1 ) ( k 1 r ) + g ν N ν ( 1 ) ( k 1 r ) ]
k 1 = 2 π / λ ɛ .
r ^ × ( H dip + H sc ) = r ^ × H tr
r ^ × ( E dip + E sc ) = r ^ × E tr
E dip = i ω ɛ 0 ɛ e × H dip
k M ν ( k r ) = × N ν ( k r )
k N ν ( k r ) = × M ν ( k r )
u ν = a n p ν
v ν = b n q ν
f ν = α n p ν
g ν = β n q ν
α n = ψ ( k a ) ζ ( k a ) ψ ( k a ) ζ ( k a ) ψ ( k 1 a ) ζ ( k a ) ψ ( k 1 a ) ζ ( k a ) / M
β n = ψ ( k a ) ζ ( k a ) ψ ( k a ) ζ ( k a ) ψ ( k 1 a ) ζ ( k a ) ψ ( k 1 a ) ζ ( k a ) / M
W RAD = ɛ 0 μ 0 π k 2 ν δ m ( 2 n + 1 ) ( n m ) ! 8 n ( n + 1 ) ( n + m ) ! [ | s ν + u ν | 2 + | t ν + v ν | 2 ]
W NON RAD = 1 2 sphere κ | E tr ( r ) | 2 d r

Metrics