Abstract

Toroidal multipoles are the subject of growing interest because of their unusual electromagnetic properties different from the electric and magnetic multipoles. In this paper, we present two new related classes of plasmonic metamaterial composed of purposely arranged of four U-shaped split ring resonators (SRRs) that show profound resonant toroidal responses at optical frequencies. The toroidal and magnetic responses were investigated by the finite-element simulations. A phenomenon of reversed toroidal responses at higher and lower resonant frequencies has also been reported between this two related metamaterials which results from the electric and magnetic dipoles interaction. Finally, we propose a physical model based on coupled LC circuits to quantitatively analyze the coupled system of the plasmonic toroidal metamaterials.

© 2012 OSA

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  1. Ia. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1184 (1958).
  2. V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990).
    [CrossRef]
  3. A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
    [CrossRef] [PubMed]
  4. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
    [CrossRef] [PubMed]
  5. B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
    [CrossRef] [PubMed]
  6. W. T. Chen, P. C. Wu, C. J. Chen, H. Y. Chung, Y. F. Chau, C. H. Kuan, and D. P. Tsai, “Electromagnetic energy vortex associated with sub-wavelength plasmonic Taiji marks,” Opt. Express 18(19), 19665–19671 (2010).
    [CrossRef] [PubMed]
  7. S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
    [CrossRef] [PubMed]
  8. D. Diessel, M. Decker, S. Linden, and M. Wegener, “Near-field optical experiments on low-symmetry split-ring-resonator arrays,” Opt. Lett. 35(21), 3661–3663 (2010).
    [CrossRef] [PubMed]
  9. W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
    [CrossRef] [PubMed]
  10. W. T. Chen, C. J. Chen, P. C. Wu, S. Sun, L. Zhou, G. Y. Guo, C. T. Hsiao, K. Y. Yang, N. I. Zheludev, and D. P. Tsai, “Optical magnetic response in three-dimensional metamaterial of upright plasmonic meta-molecules,” Opt. Express 19(13), 12837–12842 (2011).
    [CrossRef] [PubMed]
  11. S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
    [CrossRef] [PubMed]
  12. E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
    [CrossRef] [PubMed]
  13. C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
    [CrossRef] [PubMed]
  14. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
    [CrossRef] [PubMed]
  15. E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
    [CrossRef] [PubMed]
  16. V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
    [CrossRef] [PubMed]
  17. E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
    [CrossRef]
  18. T. Li, R. X. Ye, C. Li, H. Liu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Structural-configurated magnetic plasmon bands in connected ring chains,” Opt. Express 17(14), 11486–11494 (2009).
    [CrossRef] [PubMed]
  19. N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
    [CrossRef]
  20. K. Marinov, A. D. Boardman, V. A. Fedotov, and N. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
    [CrossRef]
  21. 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] [PubMed]
  22. 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] [PubMed]
  23. D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
    [CrossRef] [PubMed]
  24. J. H. Cho and D. H. Gracias, “Self-assembly of lithographically patterned nanoparticles,” Nano Lett. 9(12), 4049–4052 (2009).
    [CrossRef] [PubMed]

2011

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

W. T. Chen, C. J. Chen, P. C. Wu, S. Sun, L. Zhou, G. Y. Guo, C. T. Hsiao, K. Y. Yang, N. I. Zheludev, and D. P. Tsai, “Optical magnetic response in three-dimensional metamaterial of upright plasmonic meta-molecules,” Opt. Express 19(13), 12837–12842 (2011).
[CrossRef] [PubMed]

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

2010

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

W. T. Chen, P. C. Wu, C. J. Chen, H. Y. Chung, Y. F. Chau, C. H. Kuan, and D. P. Tsai, “Electromagnetic energy vortex associated with sub-wavelength plasmonic Taiji marks,” Opt. Express 18(19), 19665–19671 (2010).
[CrossRef] [PubMed]

D. Diessel, M. Decker, S. Linden, and M. Wegener, “Near-field optical experiments on low-symmetry split-ring-resonator arrays,” Opt. Lett. 35(21), 3661–3663 (2010).
[CrossRef] [PubMed]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

2009

J. H. Cho and D. H. Gracias, “Self-assembly of lithographically patterned nanoparticles,” Nano Lett. 9(12), 4049–4052 (2009).
[CrossRef] [PubMed]

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] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

T. Li, R. X. Ye, C. Li, H. Liu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Structural-configurated magnetic plasmon bands in connected ring chains,” Opt. Express 17(14), 11486–11494 (2009).
[CrossRef] [PubMed]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

2007

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

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

2005

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

1990

V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990).
[CrossRef]

1958

Ia. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1184 (1958).

Bitzer, A.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

Boardman, A. D.

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

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

Bourouina, T.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Brener, I.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Burckel, D. B.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Cao, J. X.

Chau, Y. F.

Chen, C. J.

Chen, W. T.

Chen, Y.

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

Cho, J. H.

J. H. Cho and D. H. Gracias, “Self-assembly of lithographically patterned nanoparticles,” Nano Lett. 9(12), 4049–4052 (2009).
[CrossRef] [PubMed]

Chong, C. T.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Chung, H. Y.

Decker, M.

Diessel, D.

Dubovik, V. M.

V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990).
[CrossRef]

Ellis, A. R.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Enkrich, C.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Fedotov, V. A.

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

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

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

Giessen, H.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Ginn, J. C.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Gracias, D. H.

J. H. Cho and D. H. Gracias, “Self-assembly of lithographically patterned nanoparticles,” Nano Lett. 9(12), 4049–4052 (2009).
[CrossRef] [PubMed]

Guo, G. Y.

Guo, H. C.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Halas, N. J.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Hsiao, C. T.

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] [PubMed]

Koschny, T.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Kuan, C. H.

Kuo, P.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

Kwong, D. L.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Li, C.

Li, J. S.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

Li, T.

Linden, S.

D. Diessel, M. Decker, S. Linden, and M. Wegener, “Near-field optical experiments on low-symmetry split-ring-resonator arrays,” Opt. Lett. 35(21), 3661–3663 (2010).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Liu, A. Q.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Liu, H.

Liu, N.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Liu, X. X.

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

Lo, G. Q.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Lu, X. C.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

Luk’yanchuk, B.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Maier, S. A.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

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] [PubMed]

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

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

Mei, T.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Nordlander, P.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Papasimakis, N.

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] [PubMed]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

Park, Y. S.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

Pendry, J. B.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Plum, E.

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Sinclair, M. B.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Smith, D. R.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

Soukoulis, C. M.

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Sun, S.

Tanaka, K.

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

Tanoto, H.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Ten Eyck, G. A.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Teng, J. H.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Tsai, D. P.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

W. T. Chen, C. J. Chen, P. C. Wu, S. Sun, L. Zhou, G. Y. Guo, C. T. Hsiao, K. Y. Yang, N. I. Zheludev, and D. P. Tsai, “Optical magnetic response in three-dimensional metamaterial of upright plasmonic meta-molecules,” Opt. Express 19(13), 12837–12842 (2011).
[CrossRef] [PubMed]

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

W. T. Chen, P. C. Wu, C. J. Chen, H. Y. Chung, Y. F. Chau, C. H. Kuan, and D. P. Tsai, “Electromagnetic energy vortex associated with sub-wavelength plasmonic Taiji marks,” Opt. Express 18(19), 19665–19671 (2010).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

Tugushev, V. V.

V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990).
[CrossRef]

Walther, M.

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

Wang, S. M.

Wegener, M.

D. Diessel, M. Decker, S. Linden, and M. Wegener, “Near-field optical experiments on low-symmetry split-ring-resonator arrays,” Opt. Lett. 35(21), 3661–3663 (2010).
[CrossRef] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Wendt, J. R.

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

Wu, P. C.

Yang, K. Y.

Ye, R. X.

Zel’dovich, Ia. B.

Ia. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1184 (1958).

Zhang, S.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

Zhang, W. L.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

Zhang, X.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

T. Li, R. X. Ye, C. Li, H. Liu, S. M. Wang, J. X. Cao, S. N. Zhu, and X. Zhang, “Structural-configurated magnetic plasmon bands in connected ring chains,” Opt. Express 17(14), 11486–11494 (2009).
[CrossRef] [PubMed]

Zhang, X. H.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Zhang, X. M.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Zheludev, N.

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

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

Zheludev, N. I.

W. T. Chen, C. J. Chen, P. C. Wu, S. Sun, L. Zhou, G. Y. Guo, C. T. Hsiao, K. Y. Yang, N. I. Zheludev, and D. P. Tsai, “Optical magnetic response in three-dimensional metamaterial of upright plasmonic meta-molecules,” Opt. Express 19(13), 12837–12842 (2011).
[CrossRef] [PubMed]

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

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] [PubMed]

E. Plum, V. A. Fedotov, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Towards the lasing spaser: controlling metamaterial optical response with semiconductor quantum dots,” Opt. Express 17(10), 8548–8551 (2009).
[CrossRef] [PubMed]

Zhou, J.

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Zhou, L.

Zhu, S.

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

Zhu, S. N.

Zhu, W. M.

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

Adv. Mater. (Deerfield Beach Fla.)

W. M. Zhu, A. Q. Liu, X. M. Zhang, D. P. Tsai, T. Bourouina, J. H. Teng, X. H. Zhang, H. C. Guo, H. Tanoto, T. Mei, G. Q. Lo, and D. L. Kwong, “Switchable magnetic metamaterials using micromachining processes,” Adv. Mater. (Deerfield Beach Fla.) 23(15), 1792–1796 (2011).
[CrossRef] [PubMed]

D. B. Burckel, J. R. Wendt, G. A. Ten Eyck, J. C. Ginn, A. R. Ellis, I. Brener, and M. B. Sinclair, “Micrometer-scale cubic unit cell 3D metamaterial layers,” Adv. Mater. (Deerfield Beach Fla.) 22(44), 5053–5057 (2010).
[CrossRef] [PubMed]

J. Opt.

E. Plum, K. Tanaka, W. T. Chen, V. A. Fedotov, D. P. Tsai, and N. I. Zheludev, “A combinatorial approach to metamaterials discovery,” J. Opt. 13(5), 055102 (2011).
[CrossRef]

Nano Lett.

J. H. Cho and D. H. Gracias, “Self-assembly of lithographically patterned nanoparticles,” Nano Lett. 9(12), 4049–4052 (2009).
[CrossRef] [PubMed]

Nat. Mater.

B. Luk’yanchuk, N. I. Zheludev, S. A. Maier, N. J. Halas, P. Nordlander, H. Giessen, and C. T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9(9), 707–715 (2010).
[CrossRef] [PubMed]

Nat. Photonics

N. Liu, H. Liu, S. Zhu, and H. Giessen, “Stereometamaterials,” Nat. Photonics 3(3), 157–162 (2009).
[CrossRef]

New J. Phys.

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

Opt. Express

Opt. Lett.

Phys. Rep.

V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990).
[CrossRef]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: finite-difference time-domain modeling,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(3 Pt 2), 036603 (2005).
[CrossRef] [PubMed]

Phys. Rev. Lett.

S. Zhang, Y. S. Park, J. S. Li, X. C. Lu, W. L. Zhang, and X. Zhang, “Negative refractive index in chiral metamaterials,” Phys. Rev. Lett. 102(2), 023901 (2009).
[CrossRef] [PubMed]

E. Plum, X. X. Liu, V. A. Fedotov, Y. Chen, D. P. Tsai, and N. I. Zheludev, “Metamaterials: optical activity without chirality,” Phys. Rev. Lett. 102(11), 113902 (2009).
[CrossRef] [PubMed]

V. A. Fedotov, N. Papasimakis, E. Plum, A. Bitzer, M. Walther, P. Kuo, D. P. Tsai, and N. I. Zheludev, “Spectral collapse in ensembles of metamolecules,” Phys. Rev. Lett. 104(22), 223901 (2010).
[CrossRef] [PubMed]

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] [PubMed]

Science

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] [PubMed]

C. M. Soukoulis, S. Linden, and M. Wegener, “Physics. Negative refractive index at optical wavelengths,” Science 315(5808), 47–49 (2007).
[CrossRef] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
[CrossRef] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[CrossRef] [PubMed]

S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306(5700), 1351–1353 (2004).
[CrossRef] [PubMed]

Sov. Phys. JETP

Ia. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1184 (1958).

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram of the toroidal metamolecules design and the polarization configuration of incident light. (a) Feature sizes of a SRRs, l = 300 nm, h1 = w1 = w2 = 50 nm, h2 = 200 nm. The U-shaped SRR play a role as ‘atom’ of toroidal metamolecule. (b) A unit cell of TM1 consisted by four-up U-shaped SRRs. (c) A unit cell of TM2 consisted by two-up-two-down U-shaped SRRs . In both Fig. 1(b) and 1(c), the periodicity ax = 1200 nm and, az = 800 nm, and the distance r from the center of toroidal metamolecule to the geometric center of SRRs is 300 nm.

Fig. 2
Fig. 2

Simulated transmittance and reflectance spectra of TM1 (a) and TM2 (d). The superscript “+” and “-” correspond to the higher and lower energy levels of each TM. The subscript “M ” and “T ” corresponding to magnetic resonance and toroidal resonance, respectively. Dispersion of radiated powers of various multipole moments for TM1 (b) and TM2 (e). Hybridization model of metamolecules for TM1 (c) and TM2 (f). The resonance energy of two-up U-shaped SRRs and two-down U-shaped SRRs are degenerated. Due to the coupling effect of SRRs, the energy level of degenerated resonance mode are split into toroidal and magnetic resonance. It’s notably that the splitting of energy level in the case of TM1 is larger than that of TM2, and the toroidal resonance of TM1 is excited at higher frequency than that of TM2.

Fig. 3
Fig. 3

Magnetic energy (color map, in logarithm scale), streamline of magnetic field (black line) and current density of z-component distributions (shown in color in second row) at respective resonances for the TM1 (a) and TM2 (b). The third row: schematic diagram of generated magnetic resonance (M) and toroidal resonance (T) in the case of TM1 and TM2. The magnetic dipole and surface current of a SRR are shown in red and blue arrow, respectively. The lowest row: schematic diagram of the electric (in blue arrow) and magnetic (in red arrow) dipoles interactions, which can explain the resonant modes in higher or lower energy.

Fig. 4
Fig. 4

Simulated transmission spectra as a function of radius r of TM1 (a) and TM2 (b). (c) The normalized coupling coefficients as a function of radius r obtained from the resonance frequencies in (a) and (b).

Equations (10)

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

electric dipole moment: P= 1 iω j d 3 r,
magnetic dipole moment: M= 1 2c ( r×j ) d 3 r,
toroidal dipole moment: T= 1 10c [ ( rj )r2 r 2 j ] d 3 r,
electric quadrupole moment: Q αβ = 1 i2ω [ r α j β + r β j α 2 3 ( rj ) δ αβ ] d 3 r,
magnetic quadrupole moment: M αβ = 1 3c [ ( r×j ) α r β + ( r×j ) β r α ] d 3 r,
I= 2 ω 4 3 c 3 | P | 2 + 2 ω 4 3 c 3 | M | 2 + 4 ω 5 3 c 4 ( PT )+ 2 ω 6 3 c 5 | T | 2 + ω 6 5 c 5 | Q αβ | 2 + ω 6 40 c 5 | M αβ | 2 +O( 1 c 5 ),
Γ= L 2 ( Q ˙ 1 2 + Q ˙ 2 2 + Q ˙ 3 2 + Q ˙ 4 2 ) L 2 ω 0 2 ( Q 1 2 + Q 2 2 + Q 3 2 + Q 4 2 ) + L' 2 ( Q ˙ 1 + Q ˙ 2 Q ˙ 3 Q ˙ 4 ) 2 M H Q ˙ 1 Q ˙ 3 M H Q ˙ 2 Q ˙ 4 M E ω 0 2 Q 1 Q 3 M E ω 0 2 Q 2 Q 4 ,
Γ=L( Q ˙ a 2 + Q ˙ b 2 )L ω 0 2 ( Q a 2 + Q b 2 )+2L' ( Q ˙ a Q ˙ b ) 2 2 M H Q ˙ a Q ˙ b 2 M E ω 0 2 Q a Q b .
{ ω M = ω 0 1+ κ E 1 κ H , with Q a = Q b ω T = ω 0 1 κ E 1+4η+ κ H , with Q a = Q b ,
{ ω M1 = ω 0 1+ κ E1 1 κ H , ω T1 + = ω 0 1 κ E1 1+4η+ κ H , for TM1 ω M2 + = ω 0 1+ κ E2 1 κ H , ω T2 = ω 0 1 κ E2 1+4η+ κ H , for TM2.

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