Abstract

Plasmonic nanoparticle clusters are widely considered experimentally and numerically. In the clusters consisting of one central particle and N satellite particles, not only the magnetic modes but also the toroidal modes can exist. Here, the eigenmodes of such clusters and the corresponding excitation efficiency under the illumination of a plane wave are studied analytically by using the eigen-decomposition method. The angular dependence of the optical response of these clusters is clearly demonstrated. The behavior of excitation efficiency is dependent on both the value and the parity of N, the number of satellite particles. Our results may provide a guide for the selective excitation of plasmonic modes in the plasmonic nanoparticle clusters.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2018 (1)

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

2017 (3)

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

L. Ge, L. Liu, S. Dai, J. Chai, Q. Song, H. Xiang, and D. Han, “Unidirectional scattering induced by the toroidal dipolar excitation in the system of plasmonic nanoparticles,” Opt. Express 25(10), 10853–10862 (2017).
[Crossref]

M. Darvishzadeh-Varcheie, C. Guclu, and F. Capolino, “Magnetic nanoantennas made of plasmonic nanoclusters for photoinduced magnetic field enhancement,” Phys. Rev. Appl. 8(2), 024033 (2017).
[Crossref]

2016 (1)

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

2015 (1)

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

2014 (1)

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

2013 (2)

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

2012 (2)

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

2011 (2)

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11(9), 3927–3934 (2011).
[Crossref]

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3(10), 4042–4059 (2011).
[Crossref]

2010 (3)

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

2009 (2)

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (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. Photonics 3(11), 654–657 (2009).
[Crossref]

2008 (2)

Q. Wu and W. Park, “Negative index materials based on metal nanoclusters,” Appl. Phys. Lett. 92(15), 153114 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

2007 (4)

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007).
[Crossref]

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101(12), 123102 (2007).
[Crossref]

2006 (1)

2005 (1)

2004 (2)

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

G. Bachelier and A. Mlayah, “Surface plasmon mediated Raman scattering in metal nanoparticles,” Phys. Rev. B 69(20), 205408 (2004).
[Crossref]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

2000 (1)

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

1995 (1)

Agasti, S. S.

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

Alivisatos, A. P.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Alù, A.

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. Photonics 3(11), 654–657 (2009).
[Crossref]

Bachelier, G.

G. Bachelier and A. Mlayah, “Surface plasmon mediated Raman scattering in metal nanoparticles,” Phys. Rev. B 69(20), 205408 (2004).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Biagioni, P.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Boardman, A. D.

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

Bohren, C. F.

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

Campione, S.

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

Capolino, F.

M. Darvishzadeh-Varcheie, C. Guclu, and F. Capolino, “Magnetic nanoantennas made of plasmonic nanoclusters for photoinduced magnetic field enhancement,” Phys. Rev. Appl. 8(2), 024033 (2017).
[Crossref]

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

Chai, J.

Chan, C. T.

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007).
[Crossref]

Chen, W. T.

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

Citrin, D. S.

Coronado, E. A.

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3(10), 4042–4059 (2011).
[Crossref]

Dai, S.

Darvishzadeh-Varcheie, M.

M. Darvishzadeh-Varcheie, C. Guclu, and F. Capolino, “Magnetic nanoantennas made of plasmonic nanoclusters for photoinduced magnetic field enhancement,” Phys. Rev. Appl. 8(2), 024033 (2017).
[Crossref]

Das, T.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

DeCrescent, R. A.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Dionne, J. A.

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11(9), 3927–3934 (2011).
[Crossref]

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

Encina, E. R.

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3(10), 4042–4059 (2011).
[Crossref]

Engheta, N.

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. Photonics 3(11), 654–657 (2009).
[Crossref]

Fedotov, V. A.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (2009).
[Crossref]

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

Fiebig, M.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Forchel, A.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Ford, G. W.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Fröhlich, D.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Fung, K. H.

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

K. H. Fung and C. T. Chan, “Plasmonic modes in periodic metal nanoparticle chains: a direct dynamic eigenmode analysis,” Opt. Lett. 32(8), 973–975 (2007).
[Crossref]

García-Etxarri, A.

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11(9), 3927–3934 (2011).
[Crossref]

Ge, L.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

L. Ge, L. Liu, S. Dai, J. Chai, Q. Song, H. Xiang, and D. Han, “Unidirectional scattering induced by the toroidal dipolar excitation in the system of plasmonic nanoparticles,” Opt. Express 25(10), 10853–10862 (2017).
[Crossref]

Geisler, P.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Giessen, H.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Guclu, C.

M. Darvishzadeh-Varcheie, C. Guclu, and F. Capolino, “Magnetic nanoantennas made of plasmonic nanoclusters for photoinduced magnetic field enhancement,” Phys. Rev. Appl. 8(2), 024033 (2017).
[Crossref]

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

Han, D.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

L. Ge, L. Liu, S. Dai, J. Chai, Q. Song, H. Xiang, and D. Han, “Unidirectional scattering induced by the toroidal dipolar excitation in the system of plasmonic nanoparticles,” Opt. Express 25(10), 10853–10862 (2017).
[Crossref]

Hecht, B.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Hentschel, M.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Hong, M.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Hopkins, B.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Hu, P.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

Huang, J. S.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Huang, Y. W.

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

Huffman, D. R.

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

Iyer, P. P.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

Jiang, T.

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

Kaelberer, T.

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

Kamp, M.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Kern, J.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Kim, C.

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

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. Photonics 3(11), 654–657 (2009).
[Crossref]

Kivshar, Y.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Kivshar, Y. S.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

Kohn, K.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Leute, S.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Li, X.

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

Liu, L.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

L. Ge, L. Liu, S. Dai, J. Chai, Q. Song, H. Xiang, and D. Han, “Unidirectional scattering induced by the toroidal dipolar excitation in the system of plasmonic nanoparticles,” Opt. Express 25(10), 10853–10862 (2017).
[Crossref]

Liu, N.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Liu, Q.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

Liu, W.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

Lottermoser, T.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Maier, S. A.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Marinov, K.

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (2009).
[Crossref]

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

Markel, V. A.

Miroshnichenko, A. E.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

Mlayah, A.

G. Bachelier and A. Mlayah, “Surface plasmon mediated Raman scattering in metal nanoparticles,” Phys. Rev. B 69(20), 205408 (2004).
[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. Photonics 3(11), 654–657 (2009).
[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. Photonics 3(11), 654–657 (2009).
[Crossref]

Neshev, D. N.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

Papasimakis, N.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (2009).
[Crossref]

Park, W.

Q. Wu and W. Park, “Negative index materials based on metal nanoclusters,” Appl. Phys. Lett. 92(15), 153114 (2008).
[Crossref]

Pavlov, V. V.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Phillips, C.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Pisarev, R. V.

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Ragan, R.

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

Rahmani, M.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Raybould, T. A.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

Rotello, V. M.

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

Saha, K.

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

Salandrino, A.

Saliba, M.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Savinov, V.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

Schuller, J. A.

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

Sheikholeslami, S. N.

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11(9), 3927–3934 (2011).
[Crossref]

Simovski, C. R.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101(12), 123102 (2007).
[Crossref]

Song, Q.

Sonnefraud, Y.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

Stefani, F. D.

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3(10), 4042–4059 (2011).
[Crossref]

Tretyakov, S. A.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101(12), 123102 (2007).
[Crossref]

Tsai, D. P.

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

Van Duyne, R. P.

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Viitanen, A. J.

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101(12), 123102 (2007).
[Crossref]

Vogelgesang, R.

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

Weber, W. H.

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

Weinmann, P.

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Willets, K. A.

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Wu, P. C.

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

Wu, Q.

Q. Wu and W. Park, “Negative index materials based on metal nanoclusters,” Appl. Phys. Lett. 92(15), 153114 (2008).
[Crossref]

Xiang, H.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

L. Ge, L. Liu, S. Dai, J. Chai, Q. Song, H. Xiang, and D. Han, “Unidirectional scattering induced by the toroidal dipolar excitation in the system of plasmonic nanoparticles,” Opt. Express 25(10), 10853–10862 (2017).
[Crossref]

Yang, W.

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

Yoxall, E.

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[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. Photonics 3(11), 654–657 (2009).
[Crossref]

Zhang, Z. Q.

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

Zheludev, N. I.

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (2009).
[Crossref]

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

ACS Nano (2)

M. Rahmani, E. Yoxall, B. Hopkins, Y. Sonnefraud, Y. Kivshar, M. Hong, C. Phillips, S. A. Maier, and A. E. Miroshnichenko, “Plasmonic Nanoclusters with Rotational Symmetry: Polarization-Invariant Far-Field Response vs Changing Near-Field Distribution,” ACS Nano 7(12), 11138–11146 (2013).
[Crossref]

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
[Crossref]

ACS Photonics (1)

S. Campione, C. Guclu, R. Ragan, and F. Capolino, “Enhanced magnetic and electric fields via Fano resonances in metasurfaces of circular clusters of plasmonic nanoparticles,” ACS Photonics 1(3), 254–260 (2014).
[Crossref]

Annu. Rev. Phys. Chem. (1)

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58(1), 267–297 (2007).
[Crossref]

Appl. Phys. Lett. (1)

Q. Wu and W. Park, “Negative index materials based on metal nanoclusters,” Appl. Phys. Lett. 92(15), 153114 (2008).
[Crossref]

Chem. Rev. (1)

K. Saha, S. S. Agasti, C. Kim, X. Li, and V. M. Rotello, “Gold nanoparticles in chemical and biological sensing,” Chem. Rev. 112(5), 2739–2779 (2012).
[Crossref]

J. Appl. Phys. (1)

C. R. Simovski, A. J. Viitanen, and S. A. Tretyakov, “Sub-wavelength resolution in linear arrays of plasmonic particles,” J. Appl. Phys. 101(12), 123102 (2007).
[Crossref]

J. Opt. Soc. Am. B (2)

J. Phys. D: Appl. Phys. (1)

L. Liu, L. Ge, P. Hu, H. Xiang, W. Yang, Q. Liu, and D. Han, “Toroidal dipolar response in plasmonic nanoparticle clusters,” J. Phys. D: Appl. Phys. 51(3), 035106 (2018).
[Crossref]

Nano Lett. (3)

M. Hentschel, M. Saliba, R. Vogelgesang, H. Giessen, A. P. Alivisatos, and N. Liu, “Transition from isolated to collective modes in plasmonic oligomers,” Nano Lett. 10(7), 2721–2726 (2010).
[Crossref]

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11(9), 3927–3934 (2011).
[Crossref]

J. S. Huang, J. Kern, P. Geisler, P. Weinmann, M. Kamp, A. Forchel, P. Biagioni, and B. Hecht, “Mode imaging and selection in strongly coupled nanoantennas,” Nano Lett. 10(6), 2105–2110 (2010).
[Crossref]

Nanoscale (1)

E. A. Coronado, E. R. Encina, and F. D. Stefani, “Optical properties of metallic nanoparticles: manipulating light, heat and forces at the nanoscale,” Nanoscale 3(10), 4042–4059 (2011).
[Crossref]

Nat. Mater. (1)

N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016).
[Crossref]

Nat. Photonics (1)

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. Photonics 3(11), 654–657 (2009).
[Crossref]

Nature (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref]

New J. Phys. (1)

K. Marinov, A. D. Boardman, V. A. Fedotov, and N. I. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. Appl. (1)

M. Darvishzadeh-Varcheie, C. Guclu, and F. Capolino, “Magnetic nanoantennas made of plasmonic nanoclusters for photoinduced magnetic field enhancement,” Phys. Rev. Appl. 8(2), 024033 (2017).
[Crossref]

Phys. Rev. B (5)

G. Bachelier and A. Mlayah, “Surface plasmon mediated Raman scattering in metal nanoparticles,” Phys. Rev. B 69(20), 205408 (2004).
[Crossref]

H. Xiang, L. Ge, L. Liu, T. Jiang, Z. Q. Zhang, C. T. Chan, and D. Han, “A minimal discrete model for toroidal moments and its experimental realization,” Phys. Rev. B 95(4), 045403 (2017).
[Crossref]

K. H. Fung and C. T. Chan, “Analytical study of the plasmonic modes of a metal nanoparticle circular array,” Phys. Rev. B 77(20), 205423 (2008).
[Crossref]

W. H. Weber and G. W. Ford, “Propagation of optical excitations by dipolar interactions in metal nanoparticle chains,” Phys. Rev. B 70(12), 125429 (2004).
[Crossref]

T. Das, P. P. Iyer, R. A. DeCrescent, and J. A. Schuller, “Beam engineering for selective and enhanced coupling to multipolar resonances,” Phys. Rev. B 92(24), 241110 (2015).
[Crossref]

Phys. Rev. Lett. (2)

N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a metamolecule: wire on a torus,” Phys. Rev. Lett. 103(9), 093901 (2009).
[Crossref]

M. Fiebig, D. Fröhlich, K. Kohn, S. Leute, T. Lottermoser, V. V. Pavlov, and R. V. Pisarev, “Determination of the magnetic symmetry of hexagonal manganites by second harmonic generation,” Phys. Rev. Lett. 84(24), 5620–5623 (2000).
[Crossref]

Sci. Rep. (1)

Y. W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3(1), 1237 (2013).
[Crossref]

Science (1)

T. Kaelberer, V. A. Fedotov, N. Papasimakis, D. P. Tsai, and N. I. Zheludev, “Toroidal dipolar response in a metamaterial,” Science 330(6010), 1510–1512 (2010).
[Crossref]

Other (1)

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

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Figures (6)

Fig. 1.
Fig. 1. Schematic view of the plasmonic nanoparticle cluster. The incident plane wave is polarized along the z axis and the incident angle ϕi is defined as the angle between k and x axis. The particle on the positive x axis is the N-th sphere and the central one is the (N + 1)-th sphere.
Fig. 2.
Fig. 2. Eigenmode analysis and optical response of the clusters with N = 2, 3, 4, 5. (a1)-(d1) are the imaginary parts of the mode polarizability Im[1/λj]. (a2)-(d2) and (a3)-(d3) are the extinction cross sections of the cluster at the incident angle of ϕi=0 and π/N, respectively. The lines correspond to the analytical results and the cyan dots represent the numerical ones for the total σe. The cross sections come from each eigenmode are plotted separately. (a4)-(d4) give the total extinction cross sections at three different incident angle for comparison.
Fig. 3.
Fig. 3. Angular dependence of the coupling efficiencies (a)-(d) and the corresponding σe (e)-(h) for the eigenmodes. The numbers indicate the indexes of the eigenmodes, j. The curves are calculated at the frequencies of corresponding resonant peaks. In (e)-(h), the sum of the σe of the degenerated eigenmodes (the j-th and (Nj)-th) are also presented and denoted by “j+(Nj)”, for example “1 + 2” in (f).
Fig. 4.
Fig. 4. Symmetry of anti-symmetrical modes with respect to the incident direction with (a) for the non-degenerate mode and (b), (c) for the degenerate modes. The red arrows indicate the direction that this mode can be excited by the external wave, while the grey arrows represent the direction that cannot be excited. The numbers in the spheres indicate the relative amplitude of the dipoles.
Fig. 5.
Fig. 5. Simulated magnetic fields (H and $|{\textbf H} |$) corresponding to the first two resonant peaks (resonance for modes with j = N and 1, respectively) of σe for the clusters with N = 2, 3. (a)-(b) and (e)-(f) corresponding to the first resonant peak while (c)-(d) and (g)-(h) corresponding to the second resonant peak. The incident angle and polarization are indicated by the arrows and dots, respectively.
Fig. 6.
Fig. 6. (a)-(b) Im(γjαj) for the cluster with N = 2 at the incident angle of 0 and π/2, respectively. (c) and (d) σe for the N-th mode (j = 2). In (b) the line corresponding to j = 1 is not plotted as cj = 0. The black dash lines indicate the zero points of the curves.

Equations (28)

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r m = R 0 cos ( 2 m π / N ) e ^ x + R 0 sin ( 2 m π / N ) e ^ y .
p m = α m ( ω ) [ E m ext + n m G ( r m r n ) p n ] ,
M ( ω ) | P = | E ext ,
| p = j 1 λ j | p j p ¯ j | E ext p ¯ j | p j .
p m j = { e i 2 π j m / N m = 1 , 2 , , N 0 m = N + 1 ,
λ j  =  α s 1 k 0 3 m = 1 N 1 A ( k 0 D m ) e i 2 π j m / N ,
p m j = { 1 m = 1 , 2 , , N p c,N ,   p c,N + 1 m = N + 1 ,
λ N , N  + 1 = ( M 11 + α c 1 ± ( M 11 α c 1 ) 2 + 4 M 12 M 21 ) / 2 ,
p m j = { cos ( 2 π j m / N ) m = 1 , 2 , , N 0 m = N + 1 ;
p m j = { i sin ( 2 π ( N j ) m / N ) m = 1 , 2 , , N 0 m = N + 1 ;
p m N / 2 = { e i m π m = 1 , 2 , , N 0 m = N + 1 .
p ¯ j | E ext = E 0 { N l = i l N J l N ( k 0 R 0 ) e i l N ϕ i + p c,N/ c,N + 1 } ,
p ¯ j | E ext = N E 0 ( J 0 ( k 0 R 0 ) + 2 i N J N ( k 0 R 0 ) cos ( N ϕ i ) ) + E 0 p c,N/ c,N + 1 .
p ¯ j | E ext = E 0 N [ i j 2 J j ( k 0 R 0 ) cos ( j ϕ i ) + i N j 2 J N j ( k 0 R 0 ) cos ( ( N j ) ϕ i ) ] ,
p ¯ j | E ext = E 0 N [ i j + 1 2 J j ( k 0 R 0 ) sin ( j ϕ i ) + i N j + 1 2 J N j ( k 0 R 0 ) sin ( ( N j ) ϕ i ) ] ,
p ¯ N / 2 | E ext = 2 E 0 N J N / 2 ( k 0 R 0 ) i N / 2 cos ( N ϕ i / 2 ) .
σ e = 4 π k | E 0 | 2 Im j = 1 N + 1 E ext | p j p ¯ j | E ext λ j p ¯ j | p j ,
σ e j = 4 π k | E 0 | 2 Im ( γ j α j | c j | 2 | E j | 2 ) .
T ( n ) = ( 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 ) n ,
t n j = { e i 2 π j n / N j = 1 , 2 , 3 , , N 1 j = N  + 1 ,
v m j = { e i 2 π j m / N m = 1,2,3, N 0 m = N  + 1 ,
v m j = { 0 m = 1,2,3, N 1 m = N  + 1 .
p m j = { e i 2 π j m / N m = 1,2,3, N 0 m = N  + 1 ,
p m j = { c j m = 1,2,3, N c j m = N  + 1 .
λ j  =  α s 1 k 0 3 m = 1 N 1 A ( k 0 D m ) e i 2 π j m / N ,
p m j = { e i 2 π j m / N m = 1 , 2 , ,   N 0 m = N + 1 ,
λ j = ( M 11 + α c 1 ± ( M 11 α c 1 ) 2 + 4 M 12 M 21 ) / 2 ,
p m j = { 1 m = 1 , 2 , ,   N p c,N , p c,N + 1 m = N + 1 ,