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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13065–13070
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Toroidal dipole response in a multifold double-ring metamaterial

Zheng-Gao Dong, Peigen Ni, Jie Zhu, Xiaobo Yin, and X. Zhang  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13065-13070 (2012)
http://dx.doi.org/10.1364/OE.20.013065


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Abstract

The toroidal response is numerically investigated in a multifold double-ring metamaterials at the antibonding magnetic-dipole mode (i.e., antiparallel magnetic dipoles in one double-ring fold). This intriguing toroidal resonance in metamaterials is considered as a result of the magnetoelectric effect due to the broken balance of the electric near-field environment. We demonstrate that the toroidal dipole response in metamaterials can improve the quality factor of the resonance spectrum. In viewing of the design flexibility on the double-ring geometry, such toroidal metamaterials will offer advantages in application potentials of toroidal dipolar moment.

© 2012 OSA

1. Introduction

Metamaterials can realize a lot of intriguing physical phenomena due to the flexibilities in designing the constitutive meta-atoms and meta-molecules by structured elements. Some of these phenomena are inexistent in naturally occurring materials, such as the left-handed electromagnetic behavior and the cloaking [1

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

,2

2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

]. Moreover, some of them are analogous to those existed in natural material systems, such as the chirality [3

3. J. Zhou, J. Dong, B. Wang, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79(12), 121104 (2009). [CrossRef]

], the electromagnetically induced transparency (EIT) [4

4. N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

], and the celestial mechanics [5

5. D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]

]. More recently, a metamaterial analogue of toroidal dipole response, represented by a closed loop of head-to-tail magnetic dipole distribution, was realized experimentally by a metamaterial composed of metallic split-ring array [6

6. 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]

,7

7. 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(2), 1760–1768 (2012). [CrossRef] [PubMed]

]. As is well known, electric and magnetic dipoles are two fundamental responses of an object in interaction with incident electromagnetic waves, corresponding to a charge oscillation and a circulating electric current, respectively. Higher-order responses, namely, multipoles like quadrupole and octupole, are usually weak as compared with the electric or magnetic dipole response. However, in contrast to these common dipole and multipole responses, a toroidal moment is not involved in the standard multipole expansion. It is produced by poloidal currents along the toroid surface, and was referred to as an anapole moment [8

8. 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]

,9

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

]. The elusive electromagnetic response of toroidal moment has attracted great attentions in nuclear, molecular, and ferroelectric physics because of its different characteristic from the fundamental electric and magnetic dipoles [10

10. W. C. Haxton, C.-P. Liu, and M. J. Ramsey-Musolf, “Nuclear anapole moments,” Phys. Rev. C Nucl. Phys. 65(4), 045502 (2002). [CrossRef]

12

12. W. C. Haxton, “Atomic parity violation and the nuclear anapole moment,” Science 275(5307), 1753–1754 (1997). [CrossRef]

]. As a result, potential applications of the toroidal response were theoretically reviewed, based on the phase transition as well as the magnetoelectric effect [13

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

]. Other applications for the toroidal response in nanostructured artificial materials have been reported recently in literatures, such as sensibility due to the high quality factor [6

6. 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]

], circular dichroism and polarization controllability due to the optical activity [8

8. 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]

], and negative refraction and backward waves due to the toroidal nature [9

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

]. Moreover, optical nonlinearity enhancement was experimentally confirmed, attributed to the toroidal susceptibility that breaks both space-inversion and time-reversal symmetry [14

14. M. Fiebig, D. Frohlich, 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] [PubMed]

].

Thereby, it is of interest to realize the toroidal moment by metamolecules, as an analogue of naturally occurring elementary particles and magnetic materials. Unfortunately, the toroidal dipole moment can neither be obtained in a metallic torus [15

15. C. M. Dutta, T. A. Ali, D. W. Brandl, T.-H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys. 129(8), 084706 (2008). [CrossRef] [PubMed]

], nor be excited magnetically in a torus-like metamaterial by straightforwardly rotating a magnetic-dipole structure [6

6. 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]

]. A typical result of such a treatment is just a linear superposition of individual magnetic dipoles, leading to a nonvanishing net magnetic dipole moment. In a recent work of ours, the dark mode of magnetic dipole resonance inherent to the split ring was verified to be excited electrically due to the asymmetric near-field environment around the split ring [16

16. Z.-G. Dong, H. Liu, M.-X. Xu, T. Li, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Role of asymmetric environment on the dark mode excitation in metamaterial analogue of electromagnetically-induced transparency,” Opt. Express 18(21), 22412–22417 (2010). [CrossRef] [PubMed]

]. From this point of view, the toroidal dipole response in metamaterials [6

6. 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]

,7

7. 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(2), 1760–1768 (2012). [CrossRef] [PubMed]

] can be explained intuitively by a magnetoelectric effect (bianisotropy) attributed to the nonuniform near-field environment for each of the four wire loops. That is, the four loop edges parallel and near to the symmetry axis of the toroidal loop array (inner edges) are electrically unbalanced, in terms of the near field distribution, regarding to the other four loop edges parallel but far away from this axis (outer edges). Consequently, there are in-phase circulating currents on the loop surfaces flowing from inner edges to their respective outer edges, or vice versa, at the dark-mode frequency, and thus the toroidal dipole response is collectively formed.

As a matter of fact, a planar double-ring structure [see Fig. 1(a)
Fig. 1 Unit-cell schematic of the toroidal metamaterial by rotating the double-ring structure with respect to the oo’ rotation axis. The outer and inner rings are shown in different colors for eye guide. (a) The planar double ring structure (1-fold). (b) The 3-fold double-ring structure with 60-degree intervals between neighboring folds.
] can produce a magnetic resonant mode with antibonding magnetic dipoles (antiparallel magnetic dipoles) in the left and right gaps [17

17. Z.-G. Dong, M.-X. Xu, S.-Y. Lei, H. Liu, T. Li, F.-M. Wang, and S.-N. Zhu, “Negative refraction with magnetic resonance in a metallic double-ring metamaterial,” Appl. Phys. Lett. 92(6), 064101 (2008). [CrossRef]

]. This antibonding magnetic-dipole mode, fully different from the routinely adopted magnetic-dipole response in a split-ring resonator, contributes to a Fano-type resonance spectrum, and has been studied in the background of negative refraction as well as EIT [17

17. Z.-G. Dong, M.-X. Xu, S.-Y. Lei, H. Liu, T. Li, F.-M. Wang, and S.-N. Zhu, “Negative refraction with magnetic resonance in a metallic double-ring metamaterial,” Appl. Phys. Lett. 92(6), 064101 (2008). [CrossRef]

20

20. J. Kim, R. Soref, and W. R. Buchwald, “Multi-peak electromagnetically induced transparency (EIT)-like transmission from bull’s-eye-shaped metamaterial,” Opt. Express 18(17), 17997–18002 (2010). [CrossRef] [PubMed]

]. In this work, by virtue of the antibonding magnetic-dipole mode, we investigate the possibility of constructing the toroidal dipole moment by rotating the planar double-ring structure into a multifold torus-like metamaterial [Fig. 1(b)].

2. Numerical model for the toroidal metamaterial

3. Results and discussions

As for the geometric configuration presented earlier, the antibonding magnetic-dipole mode is clearly shown in Figs. 2(a)
Fig. 2 Magnetic field distributions for torus-like double-ring metamaterials. All plots are in the middle yz-plane of the structure. (a) and (b) 1-fold double-ring structure. Antiparallel magnetic dipoles are resonantly confined in the double-ring gap. (c) and (d) 2-fold structure with crossed double rings. Note that the same magnetic resonant mode is shared for the crossed double-ring folds. (e) and (f) 4-fold double-ring structure with a 45-degree rotation interval. The toroidal dipole response with closed head-to-tail magnetic dipole distribution is clearly formed (Black arrows indicate the directions of the magnetic field located around the double-ring gaps).
and 2(b) around 21 GHz. Figures 2(c) and 2(d) indicate that a perpendicularly arranged double-ring structure does not introduce any irrelevant mode but just bear the same antibonding magnetic-dipole resonance. Figures 2(e) and 2(f) show an obvious formation of the toroidal dipole response in the rotated double-ring metamaterial by increasing the double-ring metamolecule to four folds. As a consequence, a nonzero toroidal dipole moment, oriented along the rotation axis of the multifold torus-like double-ring, is resulted from the closed arrangement of the eight magnetic dipoles with head-to-tail distribution along the torus meridian, though there is vanishing net magnetic dipole in the yz-plane [Fig. 2(e)].

It should be emphasized that the toroidal dipole response is unlikely induced by the H-field component of the polarized incident waves, since the vertically arranged double-ring resonator without any incident H-field component perpendicular to its plane also shows the same antibonding magnetic mode [see Fig. 2(c)]. On the contrary, it is considered that the magnetoelectric effect or so-called bianisotropy (i.e., electric coupling to magnetic resonance) is responsible for this toroidal dipole behavior. As a matter of fact, it can be confirmed from Fig. 3
Fig. 3 The magnitude distribution of the induced surface current (x-component) at the toroidal dipole resonance. Antiparallel current directions are induced for each of the inner-and-outer edge pairs of the double rings, and thus the toroidal dipole moment is collectively formed.
that there are synchronous antiparallel currents induced in each of the inner-and-outer edge pairs of the double-ring metamolecule. This current flowing mode is induced by the electric near-field unbalance between the asymmetric inner and outer edges (i.e., the outer ring is not identical to the inner ring in size) [16

16. Z.-G. Dong, H. Liu, M.-X. Xu, T. Li, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Role of asymmetric environment on the dark mode excitation in metamaterial analogue of electromagnetically-induced transparency,” Opt. Express 18(21), 22412–22417 (2010). [CrossRef] [PubMed]

,20

20. J. Kim, R. Soref, and W. R. Buchwald, “Multi-peak electromagnetically induced transparency (EIT)-like transmission from bull’s-eye-shaped metamaterial,” Opt. Express 18(17), 17997–18002 (2010). [CrossRef] [PubMed]

,23

23. N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84(15), 2943–2945 (2004). [CrossRef]

]. This kind of subradiant (or dark) mode excitation due to symmetry-broken geometries usually exhibits a Fano-type spectrum, as was found in literatures [24

24. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]

28

28. C.-Y. Chen, I.-W. Un, N.-H. Tai, and T.-J. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Express 17(17), 15372–15380 (2009). [CrossRef] [PubMed]

]. Physically, this asymmetric lineshape is caused by the two-pathway interference between the narrow resonant mode and the continuum-like spectrum of the excitation [19

19. N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94(21), 211902 (2009). [CrossRef]

,29

29. S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

], rather than an interference result of two resonant modes.

In the work by T. Kaelberer et al., it was confirmed that one of the significant characteristics in the scattering spectrum for a toroidal dipole response is its higher quality factor as compared with that for a magnetic dipole mode [6

6. 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]

]. For the Fano-type resonance spectra from the antibonding magnetic-dipole mode in double-ring metamaterials, a resonance dip is always accompanied by a resonance peak in the adjacent frequency, as shown in Fig. 4(a)
Fig. 4 (a) Transmittance spectra for multifold double-ring metamaterials, where a high quality factor (Q) is obtained for the well-shaped torus-like metamolecule (4-fold case). The incident wave propagates along the z-direction with polarized electric-field component in the x-direction, as specified in Fig. 1. (b) The scattering powers of toroidal dipole (Tx), electric dipole (Px), electric quadrupole (Qe), magnetic dipole (My), and magnetic quadrupole (Qm) calculated from the 4-fold structure. The yellow shadow is for eye guide.
. Moreover, a better shaped torus-like structure with more double-ring folds shows a steeper dip-to-peak profile, which implies a higher quality factor. As far as the transmission peak is concerned, a high quality factor up to 57.6 is obtained for the 4-fold metamolecule [Fig. 4(a)]. This should be attributed to the well-shaped toroidal dipole response, for which case the toroidally confined strong magnetic field is greatly enhanced, as shown in Fig. 2(f). As is well known, a high quality factor can improve the figure of merit in sensing the refractive-index changes of the surrounding medium [27

27. F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett. 8(11), 3983–3988 (2008). [CrossRef] [PubMed]

,28

28. C.-Y. Chen, I.-W. Un, N.-H. Tai, and T.-J. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Express 17(17), 15372–15380 (2009). [CrossRef] [PubMed]

]. However, it should be noticed that more folds of double rings will lead to larger transmission suppression due to the increased resonant loss in the multifold metal elements. In addition, the resonance frequency shows a slight shift with the increasing of the rotating folds, from around 20.0 to 23.0 GHz, which originates from the coupling effect between the double-ring folds.

To the last but not the least, although the toroidal dipole response is obtained in this multifold double-ring structure, it should be made clear that if there are any other multipoles, other than the toroidal dipole moment, contributed significantly to the transmission spectra [Fig. 4(a)]. In Fig. 4(b), the scattering powers of various multipoles are calculated by the spatial distribution of current density of an individual molecule with open boundaries [6

6. 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]

,7

7. 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(2), 1760–1768 (2012). [CrossRef] [PubMed]

]. Specifically, the x-component electric dipoles located at the upper and lower gaps of the double rings (see Fig. 1) will be involved in this resonance, but its scattering cross section [blue line in Fig. 4(b)] is about 20 times weaker than the predominant toroidal moment Tx for the 4-fold structure, at the resonant frequency of 23.0 GHz. For the scattering power by magnetic dipole My [black line in Fig. 4(b)], the closed loop distribution makes the scattering field cancel each other, and thus no obvious contributions to the far-field radiation. Moreover, the radiation powers of quadrupoles, i.e., the electric quadrupole Qe and the magnetic quadrupole Qm, are smaller in strength by 3-4 orders than that scattered by the dipoles.

4. Summary

Acknowledgments

This work was supported by the US National Science Foundation (NSF) Nanoscale Science and Engineering Center CMMI-0751621. DZG also acknowledges the National Natural Science Foundation of China (Nos. 10904012 and 11174051) and the support by Youth Research Plan from SEU.

References and links

1.

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

2.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef] [PubMed]

3.

J. Zhou, J. Dong, B. Wang, Th. Koschny, M. Kafesaki, and C. M. Soukoulis, “Negative refractive index due to chirality,” Phys. Rev. B 79(12), 121104 (2009). [CrossRef]

4.

N. Liu, L. Langguth, T. Weiss, J. Kästel, M. Fleischhauer, T. Pfau, and H. Giessen, “Plasmonic analogue of electromagnetically induced transparency at the Drude damping limit,” Nat. Mater. 8(9), 758–762 (2009). [CrossRef] [PubMed]

5.

D. A. Genov, S. Zhang, and X. Zhang, “Mimicking celestial mechanics in metamaterials,” Nat. Phys. 5(9), 687–692 (2009). [CrossRef]

6.

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]

7.

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(2), 1760–1768 (2012). [CrossRef] [PubMed]

8.

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]

9.

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

10.

W. C. Haxton, C.-P. Liu, and M. J. Ramsey-Musolf, “Nuclear anapole moments,” Phys. Rev. C Nucl. Phys. 65(4), 045502 (2002). [CrossRef]

11.

A. Ceulemans, L. F. Chibotaru, and P. W. Fowler, “Molecular anapole moments,” Phys. Rev. Lett. 80(9), 1861–1864 (1998). [CrossRef]

12.

W. C. Haxton, “Atomic parity violation and the nuclear anapole moment,” Science 275(5307), 1753–1754 (1997). [CrossRef]

13.

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

14.

M. Fiebig, D. Frohlich, 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] [PubMed]

15.

C. M. Dutta, T. A. Ali, D. W. Brandl, T.-H. Park, and P. Nordlander, “Plasmonic properties of a metallic torus,” J. Chem. Phys. 129(8), 084706 (2008). [CrossRef] [PubMed]

16.

Z.-G. Dong, H. Liu, M.-X. Xu, T. Li, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Role of asymmetric environment on the dark mode excitation in metamaterial analogue of electromagnetically-induced transparency,” Opt. Express 18(21), 22412–22417 (2010). [CrossRef] [PubMed]

17.

Z.-G. Dong, M.-X. Xu, S.-Y. Lei, H. Liu, T. Li, F.-M. Wang, and S.-N. Zhu, “Negative refraction with magnetic resonance in a metallic double-ring metamaterial,” Appl. Phys. Lett. 92(6), 064101 (2008). [CrossRef]

18.

P. Ding, E. J. Liang, L. Zhang, Q. Zhou, and Y. X. Yuan, “Antisymmetric resonant mode and negative refraction in double-ring resonators under normal-to-plane incidence,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79(1), 016604 (2009). [CrossRef] [PubMed]

19.

N. Papasimakis, Y. H. Fu, V. A. Fedotov, S. L. Prosvirnin, D. P. Tsai, and N. I. Zheludev, “Metamaterial with polarization and direction insensitive resonant transmission response mimicking electromagnetically induced transparency,” Appl. Phys. Lett. 94(21), 211902 (2009). [CrossRef]

20.

J. Kim, R. Soref, and W. R. Buchwald, “Multi-peak electromagnetically induced transparency (EIT)-like transmission from bull’s-eye-shaped metamaterial,” Opt. Express 18(17), 17997–18002 (2010). [CrossRef] [PubMed]

21.

Z.-G. Dong, H. Liu, T. Li, Z.-H. Zhu, S.-M. Wang, J.-X. Cao, S.-N. Zhu, and X. Zhang, “Optical loss compensation in a bulk left-handed metamaterial by the gain in quantum dots,” Appl. Phys. Lett. 96(4), 044104 (2010). [CrossRef]

22.

F. M. Wang, H. Liu, T. Li, Z. G. Dong, S. N. Zhu, and X. Zhang, “Metamaterial of rod pairs standing on gold plate and its negative refraction property in the far-infrared frequency regime,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(1), 016604 (2007). [CrossRef] [PubMed]

23.

N. Katsarakis, T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Electric coupling to the magnetic resonance of split ring resonators,” Appl. Phys. Lett. 84(15), 2943–2945 (2004). [CrossRef]

24.

V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef] [PubMed]

25.

R. Singh, C. Rockstuhl, F. Lederer, and W. Zhang, “Coupling between a dark and a bright eigenmode in a terahertz metamaterial,” Phys. Rev. B 79(8), 085111 (2009). [CrossRef]

26.

K. Aydin, I. M. Pryce, and H. A. Atwater, “Symmetry breaking and strong coupling in planar optical metamaterials,” Opt. Express 18(13), 13407–13417 (2010). [CrossRef] [PubMed]

27.

F. Hao, Y. Sonnefraud, P. V. Dorpe, S. A. Maier, N. J. Halas, and P. Nordlander, “Symmetry breaking in plasmonic nanocavities: subradiant LSPR sensing and a tunable Fano resonance,” Nano Lett. 8(11), 3983–3988 (2008). [CrossRef] [PubMed]

28.

C.-Y. Chen, I.-W. Un, N.-H. Tai, and T.-J. Yen, “Asymmetric coupling between subradiant and superradiant plasmonic resonances and its enhanced sensing performance,” Opt. Express 17(17), 15372–15380 (2009). [CrossRef] [PubMed]

29.

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: February 6, 2012
Revised Manuscript: April 30, 2012
Manuscript Accepted: May 21, 2012
Published: May 25, 2012

Citation
Zheng-Gao Dong, Peigen Ni, Jie Zhu, Xiaobo Yin, and X. Zhang, "Toroidal dipole response in a multifold double-ring metamaterial," Opt. Express 20, 13065-13070 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13065


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References

  1. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science305(5685), 788–792 (2004). [CrossRef] [PubMed]
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