Abstract

Due to metal losses in plasmonic metamaterials, high-refractive-index dielectrics are promising to improve optical performances of their metallic counterparts. In this paper, a LiTaO3 microtube metamaterial is numerically investigated to explore the toroidal dipolar resonance based on the multipole expansion theory. The local field strength probed on the central axis of the microtube is greatly enhanced for the toroidal dipolar mode, forming a strong hot spot concentrated in the deep-subwavelength scale. Furthermore, we also show the influences of geometrical parameter on the quality (Q) factor of the toroidal mode. The high Q factor and strongly concentrated field strength in the toroidal microtube metamaterial offer application potentials such as sensing, energy havesting, particle trapping, and nonlinear optical effects.

© 2015 Optical Society of America

1. Introduction

The classical multipole expansion primarily includes two families of multipoles, i.e., electric and magnetic dipoles, quadrupoles, octupoles, and so on [1–3], which was theoretically incomplete until the concept of toroidal multipoles was introduced to this expansion system. In 1957, Zel’dovich firstly proposed the toroidal dipole, characterized by a vortex distribution of magnetic dipoles, to explain the parity violation in the weak interaction force [4,5]. As the third family of electromagnetic multipoles, a toroidal dipole is inherently different from conventional electromagnetic dipoles (i.e., electric and magnetic dipoles) in that a toroidal dipole typically arises from a current flowing on the surface of a torus along its meridian [6–10], while an electric or a magnetic dipole is resulted from a pair of opposite charges or a current loop, respectively. Static electric polarization in natural materials can generate static toroidal dipole, which is interesting due to the intriguing feature of simultaneous violation of both space-inversion and time-reversal symmetries [1,7,11]. For such static toroidal phenomenon, there were many works focused on naturally occurring media, such as DNA condensates and multiferroics [12–15]. On the other hand, dynamic current/field may produce dynamic toroidal dipole excitation [8,16,17]. However, an optical toroidal-dipole response in natural materials, if any, is much weaker than other conventional multipolar responses in the optical range, such as electric, magnetic dipoles, or even their high-order multipoles, and thus was unfortunately out of general attentions.

Metamaterials are composed of artificially constructed, periodically arranged, sub-wavelength structures with unit size much smaller than the operating wavelength [18,19]. Most importantly of all, metamaterials own various novel electromagnetic properties unattainable in natural media, such as left-handed electromagnetic behavior [20], super imaging [21], electromagnetically induced transparency [22], perfect absorber [23], and cloaking [24]. Therefore, it has attracted extensive attentions in the science community. On the other hand, due to the subwavelength resonant feature, numerous physical effects that are normally weak in naturally occurring materials can be significantly enhanced, e.g., fluorescence, particle trapping, and the one we focused here, toroidal dipolar response. In 2007, the metamaterial of a 3D-array of toroidal solenoids was put forward by K. Marinov et al. to theoretically explore the toroidal dipolar response [6]. After that, based on the multipole expansion theory, serval toroidal metastructures [7,16,25–28] were proposed to explore the toroidal dipole resonance. For example, toroidal metamaterials of four split-ring resonators were demonstrated, from microwave [1] to optical frequencies [17], by suppressing the electric and magnetic multipolar resonances while enhancing the toroidal dipolar response. Even so, the optical ohmic-damping loss is, unfortuately, still a common issue in various plasmonic metamaterials by noble metals (Ag and Au) [7,25–27].

To avoid the loss issue in plasmonic metamaterials, all-dielectric metamaterials are competent due to their low-loss responses [29–32]. Generally, alternative dielectric materials with high refractive indices are obtainable for different frequency ranges. While a ferroelectric is of a good choice for microwave response and a semiconductor is suitable for optical range, the polaritonic LiTaO3 material is preferred to obtain a terahertz response. More recently, a metastructure consisting of four LiTaO3 solid cylinders as a cluster was investigated, where the dominant toroidal dipolar response was verified by this composite structure [8]. In this work, a simplified polaritonic LiTaO3 microtube is proposed to explore the dominant toroidal dipolar response in the terahertz regime. Inherently, the origination of toroidal response in the microtube shares a similar mutual coupling nature with the four-cylinder metastructure [8]. For the latter, the mutual coupling comes from neighboring four cylinders, while for the former, it comes from excitation of different parts of tube (thinking about that a microtube is constructed by numerous cylinders, rather than just four of them).

According to the numerical results shown in this work, a low-loss toroidal resonance can be obtained with a high quality (Q) factor as well as a strong local field enhancement in the deep-subwavelength scale. These results indicate promising application potentials in improving the sensing capability, energy havesting, particle trapping, and nonlinear optical effects, based on the high-Q-factor toroidal mode with an enhanced hot spot concentrated centrally along the axis of the microtube. Considering the extensive application backgrounds based on microtubes, such a toroidal microtube is convenient for experimental explorations with certain functional purposes of application. For fabrications, it will be complicated to drill holes in the long LiTaO3 cylinders by top-down techniques, but we expect that it may be chemically synthesized by bottom-up self-assembly method.

2. Numerical model for the microtube metamaterial

The proposed metamaterial structure comprises a periodic array of infinitely-long straight LiTaO3 microtubes as shown in Fig. 1(a). The inner and outer radii as shown in Fig. 1(b) are R1=5μm and R2=25μm, respectively. In this work, the polaritonic LiTaO3 material is considered numerically with the Lorentz-type dispersion described as [8,30,33]

ε=εω2-ωL2+iωγω2ωT2+iωγ,
where the frequency of the transverse optical phonons ωT/2π=26.7THz (i.e., phonon-polariton resonant frequency), the frequency of longitudinal optical phonons ωL/2π=46.9THz, the damping factor due to dipole relaxation γ/2π=0.94THz, and the high-frequency permittivity ε=13.4. The infinitely-long microtubes are periodically arranged with an interval of 200μm. The incident terahertz wave propagates parallel to the z-direction and has a y-polarization (along the microtube). For numerical calculations, full-wave simulations based on the finite-element method were performed [34].

 figure: Fig. 1

Fig. 1 The three-dimensional (a) and elemental top-view (b) illustrations of the infinitely-long dielectric microtube metamaterial. A probe was used to monitor the local-field enhancement at the supposed hot spot for the toroidal dipolar resonance (i.e., at the center of the microtube).

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3. Results and discussions

To analysis the far-field characteristic of the polaritonic LiTaO3 microtube, the transmittance, reflectance, and absorbance spectra are shown in Fig. 2(a). Obviously, there is a resonant dip in the transmittance spectrum at 1.33 THz, corresponding to a weak absorbance about 5.5% (due to the limited imaginary part of permittivity for the polaritonic LiTaO3 material). This dielectric-based resonant mode, so-called Mie resonance, can be verified to be a toroidal dipolar response by calculating the decomposed radiated powers according to the general multipole scattering theory [9, 17]:

electricdipolemoment:P=1iωJd3r,
magneticdipolemoment:M=12c(r×J)d3r,
toroidaldipolemoment:T=110c[(rJ)r2r2J]d3r,
electricquadrupolemoment:Qαβ=1i2ω[rαJβ+rβJα23(rJ)δαβ]d3r,
magneticquadrupolemoment:Mαβ=13c[(r×J)αrβ+(r×J)βrα]d3r,
in which c is the speed of light in the vacuum, r is the distance vector from the origin to point (x, y, z) in a Cartesian coordinate system, and α,β=x,y,z. Therefore, the decomposed far-field scattered power by these multipole moments can be written as IP=2ω4|P|2/3c3, IM=2ω4|M|2/3c3, IT=2ω6|T|2/3c5, IQe=ω6|Qαβ|2/5c5, and IQm=ω6|Mαβ|2/40c5. As shown in Fig. 2(b), the calculated scattered powers manifest that the toroidal dipolar response is the dominant resonant mode by suppressing the electric dipole and other multipolar components at a broad frequency range from 1.30 to 1.45 THz. Over this whole resonant frequency band, the decomposed power for the toroidal dipolar moment is approximately more than one order stronger than other multipole moments and four times higher than total contribution of all multipoles except the toroidal dipole itself, especially around 1.33 THz. It should be mentioned that the simplified microtube metastructure exhibits a blue shift for the toroidal response as compared with a same-size four-cylinder metastructure proposed in [8]. Specifically, for a microtube with an outer radius of 8μm and an inner radius of 3μm, the toroidal dipole resonance can be found at 4.97 THz, in contrast to the toroidal frequency at 1.89 THz in [8].

 figure: Fig. 2

Fig. 2 (a) The transmittance, reflectance, and absorbance spectra. (b) Decomposed scattered power in terms of multipoles.

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For a straightforward analysis of the origination of the toroidal dipolar response, the resonant mode is visualized in terms of the displacement current density [Fig. 3(a)]. It can be found that the antiparallel displacement currents confined to the outer and inner side-walls of the microtube are formed and thus result in a closed magnetic vortex and a centrally concentrated hot-spot field as shown in Figs. 3(b) and 3(c), respectively. This local-field distribution is a unique characteristic that confirms the toroidal dipole mode in addition to the multipolar scattered powers shown in Fig. 2(b) [8,25]. From the Ey magnitude curve probed along the x-direction diametral path [see the lower panel in Fig. 3(c)], it is obvious that the hot spot has a deep-subwavelength concentration area with a radius about 10μm, which is confined by the magnetic vortex and basically attributed to the high refractive index of the polaritonic LiTaO3 material (corresponding to a permittivity of 41.4 at the considered terahertz regime). Such a deep-subwavelength field concentration is estimated to be only one eleventh of the free-space wavelength, far beyond the one-fifth amount claimed in the toroidal four-solid-cylinder cluster [8]. Another unique property for this hot spot is its extension characteristic along the dimension of microtube axis. In addition, the electric-field amplitude probed at the central axis of the microtube [Fig. 3(d)] shows a high Q factor about 20, which can even be improved in further by modulating the inner radius of the microtube (shown below). This toroidal resonant characteristic in the microtube metastructure manifests itself in terahertz application capabilities such as energy harvesting, particle trapping, and nonlinear optical phenomena preferring a significant enhancement of local-field strength. Besides, we also verify that microtubes with finite lengths can support such toroidal response as well. For example, the toroidal dipole resonance in a 30-μm-length microtube is found at 1.40 THz, exhibiting a slight blue shift as compared to the infinite case.

 figure: Fig. 3

Fig. 3 The resonant local field distribution at 1.33 THz. (a) displacement current density J, (b) H field vector, (c) Ey field map and the probed magnitude along the x-direction diametral path, and (d) the enhanced local field Ey probed at the central axis of the microtube as a function of frequency of the excitation wave.

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In order to investigate the modulation characteristic of geometrical parameter on the toroidal resonant response, Fig. 4(a) illustrates the frequency dependence of the toroidal dipolar response on inner radius R1. For R1=0μm while keeping the outer radius R2=25μm, the toroidal dipolar resonance occurs at 1.24 THz and has relatively low Q factor and hot-spot intensity. However, with increasing the inner radius R1, the toroidal dipolar frequency will experience a blue shift. Meanwhile, it is interesting to find that the Q factor will be greatly enlarged up to 118 for R1=8μm and the probed central hot-spot intensity can be significantly enhanced as well [Fig. 4(b)]. It is noted that such a Q value is the highest one than those claimed by other toroidal metamaterials with high Q factors [1,26]. Intuitively, this improved Q factor with increasing the inner radius is a consequence of the squeezing local magnetic vortex into a thinner microtube structure. However, it does not mean that a further increase of R1 would constantly lead to enhancements of the hot-spot intensity and the Q factor. In fact, it is found numerically that features of the toroidal dipolar mode cannot be excited in a dominant sense in thin microtubes, because the magnetic vortex will not always be well behaved in a thin microtube for R1 approaching R2.

 figure: Fig. 4

Fig. 4 Influence of the inner radius (R1) of the microtube on the toroidal mode. (a) Toroidal-resonance transmission spectrum. (b) Q factor and the hot-spot intensity probed at the center of the microtube.

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4. Summary

In this paper, a polaritonic LiTaO3 microtube metamaterial has been proposed and numerically simulated to investigate the toroidal dipolar response in the terahertz frequency region. The numerical results show that a dominant toroidal dipolar response can be obtained in a broad frequency range for the simplified microtube metastructure, which is theoretically verified by the multipole scattering theory. In addition, the influences of geometrical parameter on the toroidal dipolar response in terms of the transmission spectrum, Q factor, and local-field enhancement capability were also explored. It is found that, with increasing the inner radius of microtube, the toroidal dipolar resonance not only experiences a blue shift but also obtains a high Q-factor performance. Meanwhile, a strong hot spot that extends along the axis dimension of the microtube can be squeezed into deep-subwavelength scales in the other two dimensions. These demonstrated characteristics associated with the toroidal response in the proposed microtube metamaterial can provide potential applications in sensing, energy harvesting, particle trapping, nonlinear optical effects, and so on.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NSFC) (Nos. 11174051, 11374049, and 11511140278), and Natural Science Foundation of Jiangsu Province of China (BK20131283).

References and links

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]   [PubMed]  

2. C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955). [CrossRef]  

3. 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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002). [CrossRef]   [PubMed]  

4. L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).

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

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

7. Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013). [CrossRef]  

8. 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(1), 011036 (2015). [CrossRef]  

9. J. Li, Y. Zhang, R. Jin, Q. Wang, Q. Chen, and Z. Dong, “Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection,” Opt. Lett. 39(23), 6683–6686 (2014). [CrossRef]   [PubMed]  

10. 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, 1237 (2013). [CrossRef]   [PubMed]  

11. 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), 036603 (2005). [CrossRef]   [PubMed]  

12. N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005). [CrossRef]   [PubMed]  

13. I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004). [CrossRef]   [PubMed]  

14. A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009). [CrossRef]  

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

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

17. 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]  

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

19. 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]  

20. R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012). [CrossRef]  

21. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000). [CrossRef]   [PubMed]  

22. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008). [CrossRef]   [PubMed]  

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

24. B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012). [CrossRef]   [PubMed]  

25. Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012). [CrossRef]  

26. Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012). [CrossRef]   [PubMed]  

27. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008). [CrossRef]   [PubMed]  

28. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014). [CrossRef]   [PubMed]  

29. K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004). [CrossRef]  

30. F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014). [CrossRef]  

31. Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015). [CrossRef]   [PubMed]  

32. W. Liu, J. Zhang, B. Lei, H. Hu, and A. E. Miroshnichenko, “Invisible nanowires with interfering electric and toroidal dipoles,” Opt. Lett. 40(10), 2293–2296 (2015). [CrossRef]   [PubMed]  

33. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005). [CrossRef]  

34. J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007). [CrossRef]  

References

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  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] [PubMed]
  2. C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955).
    [Crossref]
  3. 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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002).
    [Crossref] [PubMed]
  4. L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).
  5. A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (2015).
    [Crossref] [PubMed]
  6. K. Marinov, A. D. Boardman, V. A. Fedotov, and N. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
    [Crossref]
  7. Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
    [Crossref]
  8. 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(1), 011036 (2015).
    [Crossref]
  9. J. Li, Y. Zhang, R. Jin, Q. Wang, Q. Chen, and Z. Dong, “Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection,” Opt. Lett. 39(23), 6683–6686 (2014).
    [Crossref] [PubMed]
  10. 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, 1237 (2013).
    [Crossref] [PubMed]
  11. 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), 036603 (2005).
    [Crossref] [PubMed]
  12. N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005).
    [Crossref] [PubMed]
  13. I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
    [Crossref] [PubMed]
  14. A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
    [Crossref]
  15. A. Ceulemans, L. F. Chibotaru, and P. W. Fowler, “Molecular anapole moments,” Phys. Rev. Lett. 80(9), 1861–1864 (1998).
    [Crossref]
  16. V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials,” Sci. Rep. 3, 2967 (2013).
    [Crossref] [PubMed]
  17. 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]
  18. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
    [Crossref] [PubMed]
  19. 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]
  20. R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
    [Crossref]
  21. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref] [PubMed]
  22. N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
    [Crossref] [PubMed]
  23. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  24. B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
    [Crossref] [PubMed]
  25. Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
    [Crossref]
  26. Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
    [Crossref] [PubMed]
  27. N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
    [Crossref] [PubMed]
  28. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
    [Crossref] [PubMed]
  29. K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
    [Crossref]
  30. F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
    [Crossref]
  31. Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
    [Crossref] [PubMed]
  32. W. Liu, J. Zhang, B. Lei, H. Hu, and A. E. Miroshnichenko, “Invisible nanowires with interfering electric and toroidal dipoles,” Opt. Lett. 40(10), 2293–2296 (2015).
    [Crossref] [PubMed]
  33. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
    [Crossref]
  34. J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007).
    [Crossref]

2015 (4)

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(1), 011036 (2015).
[Crossref]

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

Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (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(10), 2293–2296 (2015).
[Crossref] [PubMed]

2014 (3)

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

J. Li, Y. Zhang, R. Jin, Q. Wang, Q. Chen, and Z. Dong, “Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection,” Opt. Lett. 39(23), 6683–6686 (2014).
[Crossref] [PubMed]

2013 (3)

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, 1237 (2013).
[Crossref] [PubMed]

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

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

2012 (5)

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]

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
[Crossref] [PubMed]

2010 (2)

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]

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]

2009 (1)

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

2008 (2)

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

2007 (2)

J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007).
[Crossref]

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

2006 (1)

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

2005 (3)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[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), 036603 (2005).
[Crossref] [PubMed]

N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005).
[Crossref] [PubMed]

2004 (3)

I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
[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]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
[Crossref]

2002 (1)

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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002).
[Crossref] [PubMed]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

1998 (1)

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

1958 (1)

L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).

1955 (1)

C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955).
[Crossref]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Bakker, R. M.

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

Bao, Y.

Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
[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(1), 011036 (2015).
[Crossref]

Bellaiche, L.

I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
[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), 036603 (2005).
[Crossref] [PubMed]

Briggs, D. P.

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

Ceulemans, A.

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

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

Chau, Y. F.

Chen, K.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Chen, L.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Chen, Q.

Chen, W. T.

Chibotaru, L. F.

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

Chichkov, B. N.

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

Chipouline, A.

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

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Dong, Z.

Dong, Z. G.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
[Crossref] [PubMed]

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(1), 011036 (2015).
[Crossref]

Elissalde, C.

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Evlyukhin, A. B.

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

Fang, Z.

Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
[Crossref] [PubMed]

Fedotov, V.

Fedotov, V. 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(1), 011036 (2015).
[Crossref]

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

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, 1237 (2013).
[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, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[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), 036603 (2005).
[Crossref] [PubMed]

Fowler, P. W.

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

Fu, H.

I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
[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]

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]

He, X.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Ho, Y. Z.

Hu, H.

Huang, K. C.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
[Crossref]

Huang, X. C.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Huang, Y. W.

Hud, N. V.

N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005).
[Crossref] [PubMed]

Jin, R.

Joannopoulos, J. D.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
[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] [PubMed]

Kafesaki, M.

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(1), 011036 (2015).
[Crossref]

Kivshar, Y. S.

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

Kostyuchenko, V. V.

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

Kravchenko, I. I.

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

Kuznetsov, A. I.

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

Landy, N. I.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Lei, B.

Li, C.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Li, J.

J. Li, Y. Zhang, R. Jin, Q. Wang, Q. Chen, and Z. Dong, “Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection,” Opt. Lett. 39(23), 6683–6686 (2014).
[Crossref] [PubMed]

J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007).
[Crossref]

Li, J. Q.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Li, Q.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Liu, W.

Lu, C.

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Lu, C. G.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Luk’yanchuk, B.

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

Maglione, M.

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

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.

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), 036603 (2005).
[Crossref] [PubMed]

Miroshnichenko, A. E.

A. E. Miroshnichenko, A. B. Evlyukhin, Y. F. Yu, R. M. Bakker, A. Chipouline, A. I. Kuznetsov, B. Luk’yanchuk, B. N. Chichkov, and Y. S. Kivshar, “Nonradiating anapole modes in dielectric nanoparticles,” Nat. Commun. 6, 8069 (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(10), 2293–2296 (2015).
[Crossref] [PubMed]

Mock, J. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Mounaix, P.

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Naumov, I. I.

I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
[Crossref] [PubMed]

Ni, P.

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]

Ögüt, B.

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Padilla, W. J.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[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]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Pendry, J. B.

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

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

Popkov, A. F.

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

Popov, A. I.

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

Povinelli, M. L.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
[Crossref]

Prosvirnin, S. L.

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Radescu, E. E.

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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002).
[Crossref] [PubMed]

Rho, J.

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Rogacheva, A. V.

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

Sajuyigbe, S.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

Savinov, V.

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(1), 011036 (2015).
[Crossref]

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials,” Sci. Rep. 3, 2967 (2013).
[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(2), 1760–1768 (2012).
[Crossref] [PubMed]

Schurig, D.

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

Schwartz, C.

C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955).
[Crossref]

Sigle, W.

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Smith, D. R.

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

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

Soukoulis, C. M.

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(1), 011036 (2015).
[Crossref]

Talebi, N.

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Tsai, D. P.

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

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, 1237 (2013).
[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(2), 1760–1768 (2012).
[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]

Valentine, J.

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

Vaman, G.

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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002).
[Crossref] [PubMed]

van Aken, P. A.

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Vigneras, V.

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Vilfan, I. D.

N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005).
[Crossref] [PubMed]

Vogelgesang, R.

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Wang, Q.

Wang, Y.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

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]

Wood, A.

J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007).
[Crossref]

Wu, P. C.

Yahiaoui, R.

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Yang, Y.

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

Yin, X.

Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
[Crossref] [PubMed]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Yin, X. B.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Yu, Y. F.

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

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L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).

Zhang, F. L.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

Zhang, J.

Zhang, X.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
[Crossref] [PubMed]

Zhang, Y.

Zhao, Q.

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

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), 036603 (2005).
[Crossref] [PubMed]

Zheludev, N. I.

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(1), 011036 (2015).
[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, 1237 (2013).
[Crossref] [PubMed]

V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials,” Sci. Rep. 3, 2967 (2013).
[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(2), 1760–1768 (2012).
[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]

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, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Zhu, J.

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

Z. G. Dong, P. Ni, J. Zhu, X. Yin, and X. Zhang, “Toroidal dipole response in a multifold double-ring metamaterial,” Opt. Express 20(12), 13065–13070 (2012).
[Crossref] [PubMed]

Zhu, X.

Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
[Crossref] [PubMed]

Zvezdin, A. K.

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

Annu. Rev. Biophys. Biomol. Struct. (1)

N. V. Hud and I. D. Vilfan, “Toroidal DNA condensates: Unraveling the fine structure and the role of nucleation in determining size,” Annu. Rev. Biophys. Biomol. Struct. 34(1), 295–318 (2005).
[Crossref] [PubMed]

Appl. Phys. Lett. (4)

R. Yahiaoui, U. C. Chung, C. Elissalde, M. Maglione, V. Vigneras, and P. Mounaix, “Towards left-handed metamaterials using single-size dielectric resonators: The case of TiO2-disks at millimeter wavelengths,” Appl. Phys. Lett. 101(4), 042909 (2012).
[Crossref]

Z. G. Dong, J. Zhu, J. Rho, J. Q. Li, C. Lu, X. Yin, and X. Zhang, “Optical toroidal dipolar response by an asymmetric double-bar metamaterial,” Appl. Phys. Lett. 101(14), 144105 (2012).
[Crossref]

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543 (2004).
[Crossref]

F. L. Zhang, X. C. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014).
[Crossref]

J. Sci. Comput. (1)

J. Li and A. Wood, “Finite element analysis for wave propagation in double negative metamaterials,” J. Sci. Comput. 32(2), 263–286 (2007).
[Crossref]

Nano Lett. (1)

B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible,” Nano Lett. 12(10), 5239–5244 (2012).
[Crossref] [PubMed]

Nat. Commun. (2)

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
[Crossref] [PubMed]

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

Nat. Mater. (1)

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]

Nature (1)

I. I. Naumov, L. Bellaiche, and H. Fu, “Unusual phase transitions in ferroelectric nanodisks and nanorods,” Nature 432(7018), 737–740 (2004).
[Crossref] [PubMed]

New J. Phys. (1)

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955).
[Crossref]

Phys. Rev. B (3)

Z. G. Dong, J. Zhu, X. B. Yin, J. Q. Li, C. G. Lu, and X. Zhang, “All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial,” Phys. Rev. B 87(24), 245429 (2013).
[Crossref]

A. K. Zvezdin, V. V. Kostyuchenko, A. I. Popov, A. F. Popkov, and A. Ceulemans, “Toroidal moment in the molecular magnet V15,” Phys. Rev. B 80(17), 172404 (2009).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

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), 036603 (2005).
[Crossref] [PubMed]

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 Stat. Nonlin. Soft Matter Phys. 65(4), 046609 (2002).
[Crossref] [PubMed]

Phys. Rev. Lett. (4)

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

N. I. Landy, S. Sajuyigbe, J. J. Mock, D. R. Smith, and W. J. Padilla, “Perfect metamaterial absorber,” Phys. Rev. Lett. 100(20), 207402 (2008).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

N. Papasimakis, V. A. Fedotov, N. I. Zheludev, and S. L. Prosvirnin, “Metamaterial analog of electromagnetically induced transparency,” Phys. Rev. Lett. 101(25), 253903 (2008).
[Crossref] [PubMed]

Phys. Rev. X (1)

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(1), 011036 (2015).
[Crossref]

Sci. Rep. (3)

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, 1237 (2013).
[Crossref] [PubMed]

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

Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
[Crossref] [PubMed]

Science (3)

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[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. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Sov. Phys. JETP (1)

L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).

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Figures (4)

Fig. 1
Fig. 1 The three-dimensional (a) and elemental top-view (b) illustrations of the infinitely-long dielectric microtube metamaterial. A probe was used to monitor the local-field enhancement at the supposed hot spot for the toroidal dipolar resonance (i.e., at the center of the microtube).
Fig. 2
Fig. 2 (a) The transmittance, reflectance, and absorbance spectra. (b) Decomposed scattered power in terms of multipoles.
Fig. 3
Fig. 3 The resonant local field distribution at 1.33 THz. (a) displacement current density J, (b) H field vector, (c) Ey field map and the probed magnitude along the x-direction diametral path, and (d) the enhanced local field Ey probed at the central axis of the microtube as a function of frequency of the excitation wave.
Fig. 4
Fig. 4 Influence of the inner radius (R1) of the microtube on the toroidal mode. (a) Toroidal-resonance transmission spectrum. (b) Q factor and the hot-spot intensity probed at the center of the microtube.

Equations (6)

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ε = ε ω 2 - ω L 2 + i ω γ ω 2 ω T 2 + i ω γ ,
electric dipole moment : P = 1 i ω J d 3 r,
magnetic dipole moment : M = 1 2 c ( r × J ) d 3 r,
toroidal dipole moment : T = 1 10 c [ ( r J ) r 2 r 2 J ] d 3 r,
electric quadrupole moment : Q α β = 1 i 2 ω [ r α J β + r β J α 2 3 ( r J ) δ α β ] d 3 r,
magnetic quadrupole moment : M α β = 1 3 c [ ( r × J ) α r β + ( r × J ) β r α ] d 3 r ,

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