Abstract

Direct interference between the induced magnetic and electric moments in a hybrid metal-dielectric nanodisk is demonstrated for the realization of unidirectional forward scattering in the near-infrared region. Specifically, the unidirectional forward scattering with high efficiency of the designed nanoantennas is not limited to one specific wavelength but can fit multiwavelengths. The scattering properties of the metal-dielectric hybrid nanodisk at a specific wavelength can be adjusted to meet the Kerker’s type condition. In addition, the simulated optical properties of the nanoantennas reveal that the nanodisk arranged in an array exhibits efficient absorption leading to enhanced directionality. Our results provide insights into the design and engineering of highly directive nanoantennas.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

2017 (3)

2016 (4)

F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016).
[Crossref] [PubMed]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41(21), 5011–5014 (2016).
[Crossref] [PubMed]

A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrary shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016).
[Crossref]

2015 (6)

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

Y. Yang, Q. Li, and M. Qiu, “Controlling the angular radiation of single emitters using dielectric patch nanoantennas,” Appl. Phys. Lett. 2(3), 031109 (2015).
[Crossref]

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
[Crossref]

R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, “A generalized Kerker condition for highly directive nanoantennas,” Opt. Lett. 40(11), 2645–2648 (2015).
[Crossref] [PubMed]

2014 (5)

H. Alisafaee and M. A. Fiddy, “Nanoantennas for nanodisk photovoltaics,” Appl. Phys. Lett. 105(11), 113107 (2014).
[Crossref]

W. Wan, W. Zheng, Y. Chen, and Z. Liu, “From Fano-like interference to superscattering with a single metallic nanodisk,” Nanoscale 6(15), 9093–9102 (2014).
[Crossref] [PubMed]

W. J. Luyten, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).

D. K. Gramotnev and S. I. Bozhevolnyi, “Nanofocusing of electromagnetic radiation,” Nat. Photonics 8(1), 13–22 (2014).
[Crossref]

X. Ci, B. Wu, Y. Liu, G. Chen, E. Wu, and H. Zeng, “Magnetic-based Fano resonance of hybrid silicon-gold nanocavities in the near-infrared region,” Opt. Express 22(20), 23749–23758 (2014).
[Crossref] [PubMed]

2013 (4)

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, R. F. Oulton, D. N. Neshev, O. Hess, and Y. S. Kivshar, “Scattering of core-shell nanowires with the interference of electric and magnetic resonances,” Opt. Lett. 38(14), 2621–2624 (2013).
[Crossref] [PubMed]

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

2012 (3)

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20(18), 20599–20604 (2012).
[Crossref] [PubMed]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

2009 (1)

T. Pakizeh and M. Käll, “Unidirectional Ultracompact Optical Nanoantennas,” Nano Lett. 9(6), 2343–2349 (2009).
[Crossref] [PubMed]

2008 (1)

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

1983 (1)

Alaee, R.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, “A generalized Kerker condition for highly directive nanoantennas,” Opt. Lett. 40(11), 2645–2648 (2015).
[Crossref] [PubMed]

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Albella, P.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Albooyeh, M.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

Alisafaee, H.

H. Alisafaee and M. A. Fiddy, “Nanoantennas for nanodisk photovoltaics,” Appl. Phys. Lett. 105(11), 113107 (2014).
[Crossref]

Anderton, C. R.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Artemyev, Y. A.

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

Baryshnikova, K. V.

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017).
[Crossref] [PubMed]

Belov, P. A.

Bin Hasan, S.

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Bozhevolnyi, S. I.

D. K. Gramotnev and S. I. Bozhevolnyi, “Nanofocusing of electromagnetic radiation,” Nat. Photonics 8(1), 13–22 (2014).
[Crossref]

Brener, I.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Catrysse, P. B.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

Chai, J.

Chan, C. T.

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Chen, G.

Chen, Y.

W. Wan, W. Zheng, Y. Chen, and Z. Liu, “From Fano-like interference to superscattering with a single metallic nanodisk,” Nanoscale 6(15), 9093–9102 (2014).
[Crossref] [PubMed]

Cheng, W.

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
[Crossref]

Chichkov, B. N.

A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrary shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016).
[Crossref]

Ci, X.

Dai, S.

Decker, M.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Dominguez, J.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Evlyukhin, A. B.

P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017).
[Crossref] [PubMed]

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrary shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016).
[Crossref]

Eyraud, C.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Fan, S.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

Feng, T.

Feng, Y. J.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

Fiddy, M. A.

H. Alisafaee and M. A. Fiddy, “Nanoantennas for nanodisk photovoltaics,” Appl. Phys. Lett. 105(11), 113107 (2014).
[Crossref]

Filter, R.

Fischer, T.

A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrary shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016).
[Crossref]

Fofang, N. E. T.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Froufe-Pérez, L. S.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Gao, L.

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

García-Cámara, B.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Ge, L.

Geffrin, J. M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Giles, C. L.

Gómez-Medina, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Gonzales, E.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

González, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Gramotnev, D. K.

D. K. Gramotnev and S. I. Bozhevolnyi, “Nanofocusing of electromagnetic radiation,” Nat. Photonics 8(1), 13–22 (2014).
[Crossref]

Gray, S. K.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Han, D.

Han, D. Z.

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

Hess, O.

Huebner, U.

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Jain, M.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Jiang, T.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

Käll, M.

T. Pakizeh and M. Käll, “Unidirectional Ultracompact Optical Nanoantennas,” Nano Lett. 9(6), 2343–2349 (2009).
[Crossref] [PubMed]

Karabchevsky, A.

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017).
[Crossref] [PubMed]

Kerker, M.

Kivshar, Y.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Kivshar, Y. S.

Komjani, N.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

Krasnok, A. E.

Lapin, Z.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Lederer, F.

R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, “A generalized Kerker condition for highly directive nanoantennas,” Opt. Lett. 40(11), 2645–2648 (2015).
[Crossref] [PubMed]

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Lehr, D.

Li, M. L.

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

Li, Q.

Y. Yang, Q. Li, and M. Qiu, “Controlling the angular radiation of single emitters using dielectric patch nanoantennas,” Appl. Phys. Lett. 2(3), 031109 (2015).
[Crossref]

Liang, Z.

Lin, Z. F.

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Litman, A.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Liu, L.

Liu, S.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Liu, W.

Liu, Y.

Liu, Z.

W. Wan, W. Zheng, Y. Chen, and Z. Liu, “From Fano-like interference to superscattering with a single metallic nanodisk,” Nanoscale 6(15), 9093–9102 (2014).
[Crossref] [PubMed]

Liu, Z. Z.

Luk, T. S.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Luyten, W. J.

W. J. Luyten, “Directional emission from a single plasmonic scatterer,” Nat. Commun. 5, 3250 (2014).

Maria, J.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Menzel, C.

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Miroshnichenko, A. E.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, R. F. Oulton, D. N. Neshev, O. Hess, and Y. S. Kivshar, “Scattering of core-shell nanowires with the interference of electric and magnetic resonances,” Opt. Lett. 38(14), 2621–2624 (2013).
[Crossref] [PubMed]

A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20(18), 20599–20604 (2012).
[Crossref] [PubMed]

Moreno, F.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Neshev, D. N.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, R. F. Oulton, D. N. Neshev, O. Hess, and Y. S. Kivshar, “Scattering of core-shell nanowires with the interference of electric and magnetic resonances,” Opt. Lett. 38(14), 2621–2624 (2013).
[Crossref] [PubMed]

Nieto-Vesperinas, M.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Novotny, L.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Nuzzo, R. G.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Oulton, R. F.

Pakizeh, T.

T. Pakizeh and M. Käll, “Unidirectional Ultracompact Optical Nanoantennas,” Nano Lett. 9(6), 2343–2349 (2009).
[Crossref] [PubMed]

Person, S.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Pertsch, T.

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Premaratne, M.

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
[Crossref]

Pshenay-Severin, E.

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Qin, F.

F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016).
[Crossref] [PubMed]

Qiu, M.

Y. Yang, Q. Li, and M. Qiu, “Controlling the angular radiation of single emitters using dielectric patch nanoantennas,” Appl. Phys. Lett. 2(3), 031109 (2015).
[Crossref]

Reinhardt, C.

A. B. Evlyukhin, T. Fischer, C. Reinhardt, and B. N. Chichkov, “Optical theorem and multipole scattering of light by arbitrary shaped nanoparticles,” Phys. Rev. B 94(20), 205434 (2016).
[Crossref]

Rockstuhl, C.

R. Alaee, R. Filter, D. Lehr, F. Lederer, and C. Rockstuhl, “A generalized Kerker condition for highly directive nanoantennas,” Opt. Lett. 40(11), 2645–2648 (2015).
[Crossref] [PubMed]

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

R. Alaee, C. Menzel, U. Huebner, E. Pshenay-Severin, S. Bin Hasan, T. Pertsch, C. Rockstuhl, and F. Lederer, “Deep-subwavelength plasmonic nanoresonators exploiting extreme coupling,” Nano Lett. 13(8), 3482–3486 (2013).
[Crossref] [PubMed]

Rogers, J. A.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Ruan, Z.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

Sáenz, J. J.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Shalin, A. S.

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017).
[Crossref] [PubMed]

Sikdar, D.

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
[Crossref]

Simovski, C.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

Song, Q.

Staude, I.

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

Stewart, M. E.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Sun, H. B.

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Terekhov, P. D.

P. D. Terekhov, K. V. Baryshnikova, Y. A. Artemyev, A. Karabchevsky, A. S. Shalin, and A. B. Evlyukhin, “Multipolar response of nonspherical silicon nanoparticles in the visible and near-infrared spectral ranges,” Phys. Rev. B 96(3), 035443 (2017).
[Crossref]

P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017).
[Crossref] [PubMed]

Thompson, L. B.

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
[Crossref] [PubMed]

Vaillon, R.

J. M. Geffrin, B. García-Cámara, R. Gómez-Medina, P. Albella, L. S. 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).

Verslegers, L.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

Wan, W.

W. Wan, W. Zheng, Y. Chen, and Z. Liu, “From Fano-like interference to superscattering with a single metallic nanodisk,” Nanoscale 6(15), 9093–9102 (2014).
[Crossref] [PubMed]

Wang, D. S.

Wang, S. B.

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Wicks, G.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13(4), 1806–1809 (2013).
[Crossref] [PubMed]

Wu, B.

Wu, E.

Xiang, H.

Xiao, J. J.

X. M. Zhang, Q. Zhang, S. J. Zeng, Z. Z. Liu, and J. J. Xiao, “Dual-band unidirectional forward scattering with all-dielectric hollow nanodisk in the visible,” Opt. Lett. 43(6), 1275–1278 (2018).
[Crossref] [PubMed]

F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016).
[Crossref] [PubMed]

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

Xiong, S.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

Xu, X. F.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

Xu, Y.

Yang, Y.

Y. Yang, Q. Li, and M. Qiu, “Controlling the angular radiation of single emitters using dielectric patch nanoantennas,” Appl. Phys. Lett. 2(3), 031109 (2015).
[Crossref]

Yazdi, M.

R. Alaee, M. Albooyeh, M. Yazdi, N. Komjani, C. Simovski, F. Lederer, and C. Rockstuhl, “Magnetoelectric coupling in nonidentical plasmonic nanoparticles: Theory and applications,” Phys. Rev. B Condens. Matter Mater. Phys. 91(11), 115119 (2015).
[Crossref]

Yu, Z.

L. Verslegers, Z. Yu, Z. Ruan, P. B. Catrysse, and S. Fan, “From electromagnetically induced transparency to superscattering with a single structure: a coupled-mode theory for doubly resonant structures,” Phys. Rev. Lett. 108(8), 083902 (2012).
[Crossref] [PubMed]

Zeng, H.

Zeng, S. J.

Zhang, Q.

X. M. Zhang, Q. Zhang, S. J. Zeng, Z. Z. Liu, and J. J. Xiao, “Dual-band unidirectional forward scattering with all-dielectric hollow nanodisk in the visible,” Opt. Lett. 43(6), 1275–1278 (2018).
[Crossref] [PubMed]

F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016).
[Crossref] [PubMed]

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

Zhang, W.

Zhang, X. L.

X. L. Zhang, S. B. Wang, Z. F. Lin, H. B. Sun, and C. T. Chan, “Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole,” Phys. Rev. B Condens. Matter Mater. Phys. 92, 043804 (2015).

Zhang, X. M.

Zhao, J. M.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

Zheng, W.

W. Wan, W. Zheng, Y. Chen, and Z. Liu, “From Fano-like interference to superscattering with a single metallic nanodisk,” Nanoscale 6(15), 9093–9102 (2014).
[Crossref] [PubMed]

Zhu, B.

Y. J. Feng, S. Xiong, X. F. Xu, B. Zhu, J. M. Zhao, and T. Jiang, “Dielectric multilayers for antenna and cloaking devices designed from transformation electromagnetics,” Ursi International Symposium on Electromagnetic Theory, 872–875, (2013).

ACS Nano (1)

I. Staude, A. E. Miroshnichenko, M. Decker, N. E. T. Fofang, S. Liu, E. Gonzales, J. Dominguez, T. S. Luk, D. N. Neshev, I. Brener, and Y. Kivshar, “Tailoring directional scattering through magnetic and electric resonances in subwavelength silicon nanodisk,” ACS Nano 7, 7824–7832 (2013).
[Crossref] [PubMed]

ACS Photonics (1)

Q. Zhang, J. J. Xiao, M. L. Li, D. Z. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photonics 2(1), 60–65 (2015).
[Crossref]

Appl. Phys. Lett. (2)

H. Alisafaee and M. A. Fiddy, “Nanoantennas for nanodisk photovoltaics,” Appl. Phys. Lett. 105(11), 113107 (2014).
[Crossref]

Y. Yang, Q. Li, and M. Qiu, “Controlling the angular radiation of single emitters using dielectric patch nanoantennas,” Appl. Phys. Lett. 2(3), 031109 (2015).
[Crossref]

Chem. Rev. (1)

M. E. Stewart, C. R. Anderton, L. B. Thompson, J. Maria, S. K. Gray, J. A. Rogers, and R. G. Nuzzo, “Nanostructured Plasmonic Sensors,” Chem. Rev. 108(2), 494–521 (2008).
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J. Appl. Phys. (1)

D. Sikdar, W. Cheng, and M. Premaratne, “Optically resonant magneto-electric cubic nanoantennas for ultra-directional light scattering,” J. Appl. Phys. 117(8), 083101 (2015).
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Nano Lett. (3)

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Nanoscale (1)

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Phys. Rev. B (2)

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Sci. Rep. (1)

F. Qin, Q. Zhang, and J. J. Xiao, “Sub-wavelength unidirectional antenna realized by stacked spoof localized surface plasmon resonators,” Sci. Rep. 6(1), 29773 (2016).
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Figures (11)

Fig. 1
Fig. 1 Schematic illustration of the metal-dielectric hybrid nanodisk.
Fig. 2
Fig. 2 (a) Multipole contributions to the scattering cross-section and (b) Real and imaginary components of the electric and magnetic moments of individual silicon hollow nanodisk with H = 220 nm, R = 120 nm, r = 50 nm.
Fig. 3
Fig. 3 Forward and backward scattering and forward/backward ratio scattering spectra for the individual hollow silicon nanodisk with H = 220 nm, R = 120 nm, r = 50 nm.
Fig. 4
Fig. 4 (a) Multipole contributions to the scattering cross-section and (b) Forward and backward scattering and forward/backward ratio scattering spectra of the individual Au nanodisk with H = 220 nm, r = 48 nm.
Fig. 5
Fig. 5 (a) Multipole contributions to the scattering cross-section, (b) Forward and backward scattering and forward/backward ratio scattering spectra, and (c) Scattered field pattern of core-shell nanodisk. The core nanodisk is made of gold of H = 220 nm, r1 = 50 nm, r = 50 nm and the shell is the n = 3.5 dielectric with radius R = 120 nm.
Fig. 6
Fig. 6 (a) Multipole contributions to the scattering cross-section, (b) Comparison of the total total scattering cross-section with SCS, (c) Forward and backward scattering and forward/backward ratio scattering spectra, and (d) Real and imaginary components of the electric and magnetic moments of the core-shell nanodisk. The core nanodisk is made of gold of r1 = 48 nm and the shell is the n = 3.5 dielectric with H = 220 nm, R = 120 nm, r = 50 nm.
Fig. 7
Fig. 7 Spatial profiles: (a) Elecctric field and (b) Magnetic field at the wavelengths of FS/BSmax
Fig. 8
Fig. 8 Scattered field pattern of the core-shell nanodisk with H = 220 nm, R = 120 nm, r = 50 nm, rr = 48 nm calculated at the wavelength 804 nm and 924 nm. The angle of 90° (270°) corresponds to the forward (backward) direction.
Fig. 9
Fig. 9 Forward/backward ratio scattering spectra of the Si hollow nanodisk, Au nanodisk, and core-shell nanodisk with H = 220 nm, R = 120 nm, r = 50 nm, r1 = 48 nm.
Fig. 10
Fig. 10 Far-field forward-to-backward directionality G F S / B S max for electric dipole excitation.
Fig. 11
Fig. 11 Scattering pattern by an array of nanoantennas with N = 1, 2, 4, 6.

Equations (15)

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p = P ( r ) d r
m = i ω 2 [ r × P ( r ) ] d r
T = i ω 10 [ ( 2 r 2 P ( r ) ( r P ( r ) ) r ] d r
Q ^ = Q ^ + Q ^
Q ^ = 3 [ r P ( r ) + P ( r ) r 2 3 [ r P ( r ) ] U ^ ] d r
Q ^ = 2 [ r P ( r ) ] U ^ d r
M ^ = M ^ + M ^
M ^ = ω 3 i { r × P ( r ) ] r + r [ r × P ( r ) ] } d r
M ^ = ω 3 i { r × P ( r ) ] r r [ r × P ( r ) ] } d r
I = 1 4 π ε 0 [ 2 ω 4 3 c 3 | P | 2 + 2 ω 4 3 c 3 | M | 2 + 4 ω 5 3 c 4 Im ( P T * ) + 2 ω 6 3 C 5 | T | 2 + ω 6 20 c 5 | Q e | 2 + ω 6 20 c 5 | Q m | 2 ]
C s c a = I I i n c
E f a r ( r ) = k 2 4 π ε p y e i k r r ( sin φ φ ^ + cos θ cos φ θ ^ ) Z k 2 4 π m z e i k r r ( cos θ sin φ φ ^ cos φ θ ^ ) + ...
σ B W = lim r 4 π r 2 | E f a r ( φ = 0 , θ = π ) | 2 | E i n c | 2 = k 4 4 π ε 2 | E i n c | 2 | p y ε r m z c |
p y ε r m z c = 0
p y = m z c

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