Abstract

The optical radiation of a magnetic dipole coupled to a plasmonic nanoantenna composed of stacked gold nanodisks is investigated. The nanoantenna is fed by a point magnetic dipole properly situated between two nanodisks. It is found that the magnetic dipole efficiently couples to the antenna due to the excitation of an antiphase localized surface plasmon resonance mode. This leads to a remarkable enhancement of the optical radiation at the resonant wavelength associated with the hybridized mode of the nanoantenna. By introducing a slight structural asymmetry in the nanoantenna, the optical radiation is substantially altered. Interestingly, it is shown that the omnidirectional radiation of the magnetic dipole becomes almost unidirectional, further enhanced, and accessible in three-dimensions by the asymmetric nanoantenna. Based on the classical model, quasistatic theory, and electrodynamics calculations, the electromagnetic interaction of the magnetic dipole and the nanoantennas is studied. A remarkable forward-to-backward directionality is achieved by the proposed nanoantenna at visible wavelengths.

© 2012 Optical Society of America

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2012

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

S. H. Alavi Lavasani and T. Pakizeh, “Color-switched directional ultracompact optical nanoantennas,” J. Opt. Soc. Am. B 29, 1361–1366, (2012).
[CrossRef]

2011

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

2010

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]

2009

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

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett. 9, 2343–2349 (2009).
[CrossRef]

2008

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

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

E. S. Barnard, J. S. White, A. Chandran, and M. L. Brongersma, “Spectra properties of plasmonic resonator antennas,” Opt. Express 16, 16529–16537 (2008).
[CrossRef]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Express 16, 10858–10866 (2008).
[CrossRef]

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantenna,” Phys. Rev. B 78, 195111 (2008).
[CrossRef]

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

T. Pakizeh, A. Dmitriev, M. S. Abrishamian, N. Granpayeh, and M. Käll, “Structural asymmetry and induced optical magnetism in plasmonic nanosandwiches,” J. Opt. Soc. Am. B 25, 659–667 (2008).
[CrossRef]

2007

C. M. Soukoulis, S. Linden, and M. Wegner, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

W. Cai, U. K. Chettiar, H. K. Yuan, V. C. de Silva, A. V. Kildishev, V. P. Drachev, and V. M. Shalaev, “Metamagnetics with rainbow colors,” Opt. Express 15, 3333–3341 (2007).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

2006

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

2005

J. J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005).
[CrossRef]

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

2003

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

1994

1972

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

Abrishamian, M. S.

Alavi Lavasani, S. H.

Alu, A.

Alù, A.

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantenna,” Phys. Rev. B 78, 195111 (2008).
[CrossRef]

Avlasevich, Y.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Baida, F. I.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

Balanis, C. A.

C. A. Balanis, Antenna Theory: Analysis and Design3rd ed. (Wiley, 2005).

Barnard, E. S.

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Bohren, C. F.

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

Brongersma, M. L.

Burger, S.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Burr, G. W.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

Cai, W.

Chandran, A.

Chettiar, U. K.

Christy, R. W.

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

Coronado, E.

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

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]

de Silva, V. C.

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Dmitriev, A.

Dolling, G.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Dorfmuller, J.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

Drachev, V. P.

Draine, B. T.

Dregely, D.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Eisler, H. J.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Engheta, N.

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

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantenna,” Phys. Rev. B 78, 195111 (2008).
[CrossRef]

Enkrich, C.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Fan, S.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Farahani, J. N.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

Feldmann, J.

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

Fischer, U. C.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

Flatau, P. J.

Fluckiger, P.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

Gasser, P.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

Genov, D. A.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Giannini, V.

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

Giessen, H.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

Gong, Q.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Granpayeh, N.

Greffet, J. J.

J. J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005).
[CrossRef]

Grosjean, T.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

Hagness, S. C.

C. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Halas, N. J.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (IEEE, 2001).

Hecht, B.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

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

Hou, L.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Huffman, D. R.

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

Johnson, P. B.

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

Käll, M.

Kelly, L. K.

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Kern, K.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

Kildishev, A. V.

Kinkhabwala, A.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Klar, T. A.

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

Klein, M. W.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Knight, M. W.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

Kocabas, S. E.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Koschny, T.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[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]

Kuipers, L.

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Kurzinger, K.

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

Latif, S.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Li, W.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Li, Z.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegner, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Liu, H.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Liu, J.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Liu, Y. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Liu, Z. W.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Lu, G.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Ly-Gagnon, D. S.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Martin, O. J. F.

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Miller, D. A. B.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Mivelle, M.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

Moerland, R. J.

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Moerner, W. E.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Muhlschlegel, P.

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Mullen, K.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Muskens, O. L.

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

Nichel, A.

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

Nordlander, P.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

Novotny, L.

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

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

Okyay, A. K.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Pakizeh, T.

Pavius, M.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

Pohl, D. W.

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

Prodan, E.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

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]

Radloff, C.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

Ringler, M.

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

Rivas, J. Gomez

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

Sanchez-Gil, J. A.

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

Saraswat, K. C.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Schatz, G. C.

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Schmidt, F.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Schwemer, A.

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

Segerink, F. B.

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Shalaev, V. M.

Sobhani, H.

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegner, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Stefani, F. D.

Sullivan, D. M.

D. M. Sullivan, Electromagnetic Simulation Using FDTD Method (IEEE, 2000).

Sun, C.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Taflove, C.

C. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

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]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Express 16, 10858–10866 (2008).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Tang, L.

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Taubert, R.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

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]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna,” Opt. Express 16, 10858–10866 (2008).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Vogelgesang, R.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[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]

Wegener, M.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Wegner, M.

C. M. Soukoulis, S. Linden, and M. Wegner, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

White, J. S.

Wu, D. M.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Wunderlich, M.

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

Yu, Z.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

Yuan, H. K.

Yue, S.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Zhang, T.

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

Zhang, X.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

Zhao, L. L.

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Zhou, J.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

Zhu, S. N.

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

ACS Nano

G. Lu, W. Li, T. Zhang, S. Yue, J. Liu, L. Hou, Z. Li, and Q. Gong, “Plasmonic-enhanced molecular fluorescence within isolated bowtie nano-apertures,” ACS Nano 6, 1438–1448 (2012).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. Zhou, T. Koschny, C. M. Soukoulis, S. Burger, F. Schmidt, and M. Wegener, “Photonic Metamaterials: Magnetism at optical frequencies,” IEEE J. Sel. Top. Quantum Electron. 12, 1097–1105(2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. Chem. B

L. K. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[CrossRef]

Nano Lett.

T. Grosjean, M. Mivelle, F. I. Baida, G. W. Burr, and U. C. Fischer, “Diabolo nanoantenna for enhancing and confining the magnetic optical field,” Nano Lett. 11, 1009–1013 (2011).
[CrossRef]

O. L. Muskens, V. Giannini, J. A. Sanchez-Gil, and J. Gomez Rivas, “Strong enhancement of the radiative decay rate of emitters by single plasmonic nanoantennas,” Nano Lett. 7, 2871–2875 (2007).
[CrossRef]

T. Pakizeh and M. Käll, “Unidirectional ultracompact optical nanoantennas,” Nano Lett. 9, 2343–2349 (2009).
[CrossRef]

T. H. Taminiau, R. J. Moerland, F. B. Segerink, L. Kuipers, and N. F. van Hulst, “Resonance of an optical monopole antenna probed by single molecule fluorescence,” Nano Lett. 7, 28–33 (2007).
[CrossRef]

Nanotechnology

J. N. Farahani, H. J. Eisler, D. W. Pohl, M. Pavius, P. Fluckiger, P. Gasser, and B. Hecht, “Bow-tie optical antenna probes for single-emitter scanning near-field optical microscopy,” Nanotechnology 18, 125506 (2007).
[CrossRef]

Nat Commun.

D. Dregely, R. Taubert, J. Dorfmuller, R. Vogelgesang, K. Kern, and H. Giessen, “3D optical Yagi-Uda nanoantenna array,” Nat Commun. 2, 267 (2011).
[CrossRef]

Nat. Photon.

A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Mullen, and W. E. Moerner, “Large single-molecule fluorescence enhancements produced by a bowtie nanoantenna,” Nat. Photon. 3, 654–657 (2009).
[CrossRef]

L. Tang, S. E. Kocabas, S. Latif, A. K. Okyay, D. S. Ly-Gagnon, K. C. Saraswat, and D. A. B. Miller, “Nanometer-scale germanium photodetector enhanced by a near-infrared dipole antenna,” Nat. Photon. 2, 226–229 (2008).
[CrossRef]

Nature

L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008).
[CrossRef]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Opt. Express

Phys. Rev. B

H. Liu, D. A. Genov, D. M. Wu, Y. M. Liu, Z. W. Liu, C. Sun, S. N. Zhu, and X. Zhang, “Magnetic plasmon hybridization and optical activity at optical frequencies in metallic nanostructures,” Phys. Rev. B 76, 073101 (2007).
[CrossRef]

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

A. Alù and N. Engheta, “Hertzian plasmonic nanodimer as an efficient optical nanoantenna,” Phys. Rev. B 78, 195111 (2008).
[CrossRef]

Phys. Rev. Lett.

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

Science

J. J. Greffet, “Nanoantennas for light emission,” Science 308, 1561–1563 (2005).
[CrossRef]

P. Muhlschlegel, H. J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science 308, 1607–1609 (2005).
[CrossRef]

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003).
[CrossRef]

M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science 332, 702–704 (2011).
[CrossRef]

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]

C. M. Soukoulis, S. Linden, and M. Wegner, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

Other

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

C. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

D. M. Sullivan, Electromagnetic Simulation Using FDTD Method (IEEE, 2000).

R. F. Harrington, Time-Harmonic Electromagnetic Fields (IEEE, 2001).

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

C. A. Balanis, Antenna Theory: Analysis and Design3rd ed. (Wiley, 2005).

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Figures (6)

Fig. 1.
Fig. 1.

(a) Radiation patterns (red, θ ; blue, ϕ ) of a single MPS ( M x ) in free space; the normalized optical radiation patterns of (b) the symmetric nanoantenna and (c) the asymmetric nanoantenna, fed by the MPS, at wavelength λ = 625 nm . Diameter D = 88 nm and thickness T = 20 nm in (b); and in (c), diameters D 1 = 80 and D 2 = 96 nm and thickness T = 20 nm . The separation distance d = 20 nm is kept the same in (b) and (c). Clearly, the optical radiation of the MPS in (c) is almost unidirectional, compared to the bidirectional radiation in (b).

Fig. 2.
Fig. 2.

Optical radiation patterns of the asymmetric nanoantenna with the same parameters as in Fig. 1(c) at wavelength λ = 613 nm .

Fig. 3.
Fig. 3.

Spectra of the calculated far-field radiations of the MPS ( M x ) coupled to the (a) symmetric and (b) asymmetric nanoantennas, toward the FW (solid curve) and BW directions (squares), compared to their reciprocal cases, i.e., the near-field magnetic-field intensity enhancement, | H x | 2 , under the PW illuminations from the FW or BW direction; theoretical results for the confined | H x | 2 in the center of the (c) symmetric and (d) asymmetric nanoantennas.

Fig. 4.
Fig. 4.

Spectra of the calculated farfield G FB (circles) and nearfield K FB (solid curve) directionalities of the asymmetric nanoantenna, compared to the theoretical K FB obtained based on the localized H x , m .

Fig. 5.
Fig. 5.

(a), (b) Farfield results: optical extinction spectra of the asymmetric (solid curve) and symmetric (dashed curve) nanoantennas, obtained using (a) the FDTD calculations and (b) the theory. The optical extinction spectra of the single nanodisk ( D = 88 and T = 20 nm ) and the approximated nanospheroid ( 2 a = 2 b = 88 and 2 c = 20 nm ) are shown by the dotted curves in (a) and (b). (c), (d) Near-field results for the asymmetric case: (c) the numerically calculated confined E -field, | E z | , close to the two nanoparticles, upon the PW illuminations; (d) the theoretical results for the normalized modified polarizabilities, | α ˜ z , j | n , of the two interacting nanoparticles, obtained using the modified CDA method.

Fig. 6.
Fig. 6.

Intensities of the induced magnetic field ( | H x | 2 ) in the near-field of the nanoantenna, with D 1 = 96 , D 2 = 80 , and T = d = 20 nm , under the plane-wave (PW) illuminations from the FW direction [(a), (c)] and BW direction [(b), (d)], at two resonance wavelengths: λ = 613 nm [(a), (b)] and λ = 567 nm [(c), (d)].

Equations (5)

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α ( ω ) = V ( ε ε m ) [ ε m + L ( ε ε m ) ] ,
ε ( ω ) = 1 ω p 2 ω ( ω + i γ ) + j = 1 N G j ω j 2 ω j 2 ω 2 i Γ j ω ,
G FB = 10 × log 10 ( S FW / S BW ) ,
K FB 10 × log 10 [ | H x , m | FW 2 / | H x , m | BW 2 ] ,
E θ = η M x 4 π [ k 2 r + i k r 2 ] e i k r sin ( ϕ ) , H ϕ = M x 4 π [ k 2 r i k r 2 + 1 r 3 ] e i k r sin ( ϕ ) , H r = M x 2 π [ i k r 2 + 1 r 3 ] e i k r cos ( ϕ ) ,

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