Abstract

We revisit the fundamental topic of light scattering by single homogenous nanoparticles from the new perspective of excitation and manipulation of toroidal dipoles. It is revealed that besides within all-dielectric particles, toroidal dipoles can also be efficiently excited within homogenous metallic nanoparticles. Moreover, we show that those toroidal dipoles excited can be spectrally tuned through adjusting the radial anisotropy parameters of the materials, which paves the way for further more flexible manipulations of the toroidal responses within photonic systems. The study into toroidal multipole excitation and tuning within nanoparticles deepens our understanding of the seminal problem of light scattering, and may incubate many scattering related fundamental researches and applications.

© 2015 Optical Society of America

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2015 (6)

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5, 011036 (2015).

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Lukyanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref] [PubMed]

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domłnguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Lukyanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015).
[Crossref] [PubMed]

W. Liu, J. Zhang, B. Lei, H. Hu, and A. E. Miroshnichenko, “Invisible nanowires with interfering electric and toroidal dipoles,” Opt. Lett. 40, 2293–2296 (2015).
[Crossref]

W. Liu, R. F. Oulton, and Y. S. Kivshar, “Geometric interpretations for resonances of plasmonic nanoparticles,” Sci. Rep. 5, 12148 (2015).
[Crossref] [PubMed]

W. Liu, “Ultra-directional super-scattering of homogenous spherical particles with radial anisotropy,” Opt. Express 23, 14734–14743 (2015).
[Crossref] [PubMed]

2014 (6)

W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22, 16178–16187 (2014).
[Crossref] [PubMed]

L. Lu, J. D. Joannopoulos, and M. Soljacic, “Topological photonics,” Nat. Photon. 8, 821–829 (2014).
[Crossref]

F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photon. 8, 899–907 (2014).
[Crossref]

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23, 047806 (2014).
[Crossref]

V. Savinov, V. Fedotov, and N. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89, 205112 (2014).
[Crossref]

Q. Zhang, J. J. Xiao, X. M. Zhang, D. Han, and L. Gao, “Core-shell-structured dielectric-metal circular nanodisk antenna: Gap plasmon assisted magnetic toroid-like cavity modes,” ACS Photon. 2, 60–65 (2014).
[Crossref]

2013 (3)

V. A. Fedotov, A. Rogacheva, V. Savinov, D. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials,” Sci. Rep. 3, 2967 (2013).
[Crossref] [PubMed]

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Low-loss and high-q planar metamaterial with toroidal moment,” Phys. Rev. B 87, 115417 (2013).
[Crossref]

Y. X. Ni, L. Gao, A. E. Miroshnichenko, and C. W. Qiu, “Controlling light scattering and polarization by spherical particles with radial anisotropy,” Opt. Express 21, 8091–8100 (2013).
[Crossref] [PubMed]

2012 (6)

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

Z. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20, 13065–13070 (2012).
[Crossref] [PubMed]

Y.-W. Huang, W. T. Chen, P. C. Wu, V. Fedotov, V. Savinov, Y. Z. Ho, Y.-F. Chau, N. I. Zheludev, and D. P. Tsai, “Design of plasmonic toroidal metamaterials at optical frequencies,” Opt. Express 20, 1760–1768 (2012).
[Crossref] [PubMed]

B. Ogut, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12, 5239–5244 (2012).
[Crossref] [PubMed]

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. B. Zhang, and B. S. Lukyanchuk, “Magnetic light,” Sci. Rep. 2, 492 (2012).
[Crossref] [PubMed]

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

2011 (1)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photon. 5, 83–90 (2011).
[Crossref]

2010 (3)

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

C. Qiu, L. Gao, J. D. Joannopoulos, and M. Soljačić, “Light scattering from anisotropic particles: propagation, localization, and nonlinearity,” Laser Photon. Rev. 4, 268–282 (2010).
[Crossref]

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

2009 (3)

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref] [PubMed]

A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy, and A. V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. 8, 867–871 (2009).
[Crossref] [PubMed]

2007 (1)

C.-W. Qiu, L.-W. Li, T.-S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
[Crossref]

2006 (1)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[Crossref]

2002 (2)

E. E. Radescu and G. Vaman, “Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles,” Phys. Rev. E 65, 046609 (2002).
[Crossref]

E. A. Marengo and R. W. Ziolkowski, “Nonradiating sources, the aharonovbohm effect, and the question of measurability of electromagnetic potentials,” Radio Sci. 37, 19 (2002).
[Crossref]

1995 (1)

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic field of elementary time-dependent toroidal sources,” J. Phys. A Math. Gen. 28, 4565 (1995).
[Crossref]

1992 (1)

1990 (1)

V. M. Dubovik and V. V. Tugushev, “Toroidmoments in electrodynamics and solid-state physics,” Phys. Rep. 187, 145–202 (1990).
[Crossref]

1989 (1)

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852 (1989).
[Crossref]

1972 (1)

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

Afanasiev, G. N.

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic field of elementary time-dependent toroidal sources,” J. Phys. A Math. Gen. 28, 4565 (1995).
[Crossref]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73, 045105 (2006).
[Crossref]

Atkinson, R.

A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy, and A. V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. 8, 867–871 (2009).
[Crossref] [PubMed]

Atwater, H. A.

H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9, 205–213 (2010).
[Crossref] [PubMed]

Bakker, R.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref] [PubMed]

Bakker, R. M.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Lukyanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref] [PubMed]

R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domłnguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Lukyanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015).
[Crossref] [PubMed]

Bartal, G.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

Basharin, A. A.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5, 011036 (2015).

Belgrave, A. M.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser,” Nature 460, 1110–1112 (2009).
[Crossref] [PubMed]

Bohren, C. F.

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

Bozhevolnyi, S. I.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

Chau, Y.-F.

Chen, H.

Y. Fan, Z. Wei, H. Li, H. Chen, and C. M. Soukoulis, “Low-loss and high-q planar metamaterial with toroidal moment,” Phys. Rev. B 87, 115417 (2013).
[Crossref]

Chen, W. T.

Chichkov, B. N.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Lukyanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref] [PubMed]

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

Chipouline, A.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Lukyanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref] [PubMed]

Christy, R. W.

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

Dai, L.

R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
[Crossref] [PubMed]

Dong, Z.

Doyle, W. T.

W. T. Doyle, “Optical properties of a suspension of metal spheres,” Phys. Rev. B 39, 9852 (1989).
[Crossref]

Dubey, M.

F. Xia, H. Wang, D. Xiao, M. Dubey, and A. Ramasubramaniam, “Two-dimensional material nanophotonics,” Nat. Photon. 8, 899–907 (2014).
[Crossref]

Dubovik, V. M.

V. M. Dubovik and V. V. Tugushev, “Toroidmoments in electrodynamics and solid-state physics,” Phys. Rep. 187, 145–202 (1990).
[Crossref]

Economou, E. N.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5, 011036 (2015).

Eriksen, R. L.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012).
[Crossref] [PubMed]

Evans, P.

A. V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G. A. Wurtz, R. Atkinson, R. Pollard, V. A. Podolskiy, and A. V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nat. Mater. 8, 867–871 (2009).
[Crossref] [PubMed]

Evlyukhin, A. B.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Lukyanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
[Crossref] [PubMed]

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Nat. Commun. (1)

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R. F. Oulton, V. J. Sorger, T. Zentgraf, R. M. Ma, C. Gladden, L. Dai, G. Bartal, and X. Zhang, “Plasmon lasers at deep subwavelength scale,” Nature 461, 629–632 (2009).
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C.-W. Qiu, L.-W. Li, T.-S. Yeo, and S. Zouhdi, “Scattering by rotationally symmetric anisotropic spheres: Potential formulation and parametric studies,” Phys. Rev. E 75, 026609 (2007).
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V. A. Fedotov, A. Rogacheva, V. Savinov, D. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials,” Sci. Rep. 3, 2967 (2013).
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Figures (4)

Fig. 1
Fig. 1 ((a) Schematic illustration of the scattering of an incident plane wave by a spherical particle of radius R. The spherical particle can be homogenous and isotropic with reflective index n, or can be radially anisotropic with radial permittivity εr and transverse permittivity εt = n2. The anisotropy parameter is defined as η = εtr (η = 1 for isotropic spheres). The plane wave is polarized (in terms of electric field) along x direction and is propagating along z direction. (b) Schematic illustration of the toroidal dipole excitation within the spherical particle. Both the current J and magnetic field H distributions have been shown.
Fig. 2
Fig. 2 (a) Scattered power spectra for a dielectric (n = 3) sphere (inset) of R = 200 nm. The contributions from current-integrated [based on Eq.(1) and Eq.(3)] electric dipole (P, red curves) and TD (T, blue curves), and those from the far-field deduced dipole [P(a1), dashed black curves] are shown. Four points of different α = kR are indicated by black dots in (a) (A: α = 0.8, B: α = 1.5, C: α = 2.6, D: α = 3). The corresponding near fields on the xz plane of y = 0 are shown respectively in (A)-(D) in the bottom row. The distributions for both the normalized magnetic field along y direction H y (color-plots) and displacement field D = εE on plane (vector-plots, only the field inside the particle is plotted) are shown (the dashed black lines denote the boundaries of the particles), as is also the case for Figs. 34.
Fig. 3
Fig. 3 (a) Scattered power spectra for a silver sphere (inset) of R = 200 nm. The contributions from P, T, and P(a1) are shown. Two points of different α are indicated by black dots in (a) (A: α = 0.8 and B: α = 3.5). The corresponding near fields are shown in (A)–(B) on the right.
Fig. 4
Fig. 4 Scattered power spectra for dielectric spheres (inset) of radius R = 200 nm with anisotropy parameters η =εtr = 1/3 in (a) and η = 2 in (b). The transverse permittivity is fixed at εt = 9. The contributions from P, T, and P(a1) are shown. Four points are indicated by black dots in (a) and (b) (A: α = 1.27, B: α = 2.31, C: α = 1.72, D: α = 2.8). The corresponding near fields are shown respectively in (A)-(D) on the right.

Equations (9)

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P = 1 i ω d 3 r J ( r ) ,
M = 1 2 c d 3 r [ r × J ( r ) ] ,
T = 1 10 c d 3 r [ ( r J ( r ) ) r 2 r 2 J ] .
W P = μ 0 ω 4 12 π c | P | 2 , W T = μ 0 ω 4 k 2 12 π c | T | 2 ,
E s ( r ) = m = 1 E m [ i a m N m ( r ) b m M m ( r ) ] ,
P ( a 1 ) = ε 0 6 π i a 1 k 3 E 0 ,
W P ( a 1 ) = 3 π | E 0 | 2 k ω μ 0 | a 1 | 2 ,
J ( r ) = i ω ε 0 [ ε ( r ) 1 ] E ( r ) ,
P ( a 1 ) = P + i k T + ,

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