Abstract

Toroidal dipolar response in a metallic metastructure, composed of double flat rings, is utilized to manipulate the radiation pattern of a single dipolar emitter (e.g., florescent molecule/atom or quantum dot). Strong Fano-type radiation spectrum can be obtained when these two coupling dipoles are spatially overlapped, leading to significant radiation suppression (so-called nonradiating source) attributed to the dipolar destructive interference. Moreover, this nonradiating configuration will become a directionally super-radiating nanoantenna after a radial displacement of the emitter with respect to the toroidal flat-ring geometry, which emits linearly polarized radiation with orders of power enhancement in a particular orientation. The demonstrated radiation characteristics from a toroidal-dipole-mediated dipolar emitter indicate a promising manipulation capability of the dipolar emission source by intriguing toroidal dipolar response.

© 2015 Optical Society of America

1. Introduction

The toroidal moment, firstly proposed by Zel’dovich in 1957 to interpret the parity violation on the weak interaction in nuclear and particle physics [1], fertilizes the area of classical but incomplete electromagnetic theory of multipole decomposition [2–4 ]. Unlike electric and magnetic dipoles resulting from a pair of charges and circulating currents respectively, the intriguing toroidal dipole is caused by a current flowing on the surface of a torus along its meridian [5,6 ]. However, the toroidal dipolar response is much weaker than conventional dipoles and, consequently, difficult to be detected in naturally occurring media [7]. Therefore, to find dominant toroidal dipolar responses by artificial composites is an emerging exploration field. Recently, it was claimed that toroidal metamaterials could be constructed to strengthen the toroidal dipolar response while suppressing other electric and magnetic multipoles [2,5,6,8–13 ]. Though it is well known that a toroidal dipole has a radiation pattern identical to an electric or magnetic dipole [14], an interesting phenomenon associated with the toroidal dipole is the nonradiating characteristic when it interacts with an electric dipole, which can be regarded as a nontrivial anapole [2,5,12,15–18 ]. It has been numerically demonstrated that such a hypothetical nonradiating system can be converted into a directional emission source in dependence of the interface contrast of index of refraction between the dipolar system and surroundings [19]. In addition, it should be interesting to explore the interaction phenomena between a toroidal dipole and other multipoles. For example, it was reported that the toroidal dipolar response could be enhanced by coupling to electric or magnetic dipole, resulting in an analogy of electromagnetically-induced transparency [5,12,15,18 ].

In this paper, a toroidal metastructure by double metallic flat-rings is proposed to numerically investigate the scattering characteristic of its interaction with a dipolar emitter. It is interesting to find that the interaction can be either radiating enhanced or suppressed (i.e., nonradiating), depending on constructive or destructive interference, respectively. Moreover, the nonradiating phenomenon can be sensitively manipulated to be a super-radiating one with both pronounced directionality and linear polarization conversion, once the dipole emitter is shifted away from the axis of toroidal dipole.

2. Numerical model

The passive composite metastructure proposed in this paper is shown in Fig. 1 , comprising double metallic flat-ring elements for the toroidal dipolar response with a silicon-oxide gap layer as spacer (εSiO2=2.4). The flat-ring inner and outer radii are 100 and 300 nm, respectively, and the gap between 40-nm-thick metallic flat rings is 30 nm. This double metallic flat-ring metastructure has been demonstrated to support the dipolar toroidal response by using the multipole decomposition theory in our previous works [20,21 ]. In this work, an active dipole emitter is considered as a light source to excite the toroidal dipolar response. The dipolar emitter, oriented along the y-axis, with a moment μ=jI0l/ω is numerically modeled by a tiny line (l=1nm) carrying an electric current I0=1A. The metallic metastructure is considered as silver following the Drude-type dispersion model ε(ω)=ε-ωp2(ω2+iωγ), with a high-frequency permittivity ε=6.0, plasma frequency ωp=1.5×1016s1, and collision frequency γ=7.73×1013s1 [20,22,23 ]. To explore the coupling effect between toroidal dipole response and electric-dipole emitter, variable displacements of the emitter with respect to the geometric center of double flat rings are considered [i.e., axial displacement along the y axis (Δy) and radial displacement along the x axis (Δx)]. Numerical simulations are performed by a full-wave solver based on finite-difference time-domain (FDTD) method [19,20,24 ]. Far-field emission powers after scattered by the flat-ring metastructure are monitored at three coordinate directions. By excluding negligible components of radiation field, only y-component radiation power of the X-direction emission (|EyX|2), x-component radiation power of the Y-direction emission (|ExY|2), and y-component radiation power of the Z-direction emission (|EyZ|2) will be presented in this paper, corresponding to far-field probes A, B, and C, respectively (Fig. 1).

 

Fig. 1 Schematic of the metastructure composed of double metallic flat-rings with a silicon-oxide gap spacer. Note that the origin “O” of the coordinate system is located at the center of metastructure.

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3. Results and discussions

As is well-known, either a toroidal or an electric dipole exhibits a donut-like radiation pattern [14]. Therefore, for the axially symmetric modeling configuration with an on-axis displacement Δy for the emitter (Fig. 1), the radiating characteristic can be evaluated by a radially positioned far-field monitor (e.g., probe A or C). When the dipole emitter is put away from the center of the double flat rings (for example, Δy=150nm), the probed far-field power has only a slight dip at 1050 nm due to weak coupling to the toroidal response in metallic flat rings. However, when the emitter approaches, the scattering field at the toroidal resonance will exhibit a Fano-type asymmetric line shape, which is a result of strong interference between the narrow toroidal dipolar response and the broad electric dipolar source. If the dipolar emitter is exactly overlapped with the toroidal dipole (i.e., Δy=0nm), the radiation field at 1050 nm reaches a resonant peak, implying a pronounced constructive interference between toroidal and electric dipoles. In contrast, destructive interference happens at the adjacent wavelength of 945 nm, leading to a significantly suppressed radiation. The dipolar radiating enhancement peak under the assistance of a toroidal flat-ring structure is beneficial in identification of a single nano-emitter through the radiation intensity. On the other hand, the radiation suppression implies that this composite nano-system can be regarded as a so-called nonradiating source. In literatures, this kind of nonradiating system of interacting toroidal and electric dipoles was known as an anapole [14,16,18 ]. Here, because the realistic metal losses in plasmonic flat-rings results in an incompetent strength of the induced toroidal dipole as compared to the active emitter, we shall not expect a suppression of radiation with a complete destructive interference better than the calculated nonvanishing radiation dip at 945 nm.

To show a straightforward picture of the interference interaction between dipoles, local magnetic fields for the resonant radiation dip and peak are presented in Figs. 2(b) and 2(c) , respectively. In both cases, magnetic-vortex distributions as the unique feature of a dipolar toroidal mode are clearly induced by the dipole emitter, but the underlying interference interactions are not in a same way. For the resonant dip at 945 nm, the toroidal dipole mode shows a 180-degree phase lag [Fig. 2(b)], which results in a destructive interference with the dipolar excitation field and, consequently, the pronounced radiation suppression (nonradiating phenomenon). On the contrary, the toroidal mode at 1050 nm is induced in phase with the dipolar excitation field, so that the constructive interference between them causes a radiating enhancement. Besides, the calculated scattered powers in terms of multipoles also sufficiently justify the excitation of a dominant toroidal dipole around 1050 nm in Fig. 2(d).

 

Fig. 2 (a) Far-field radiating power for different axial displacements Δy. The nonradiating dips and super-radiative peaks in the Fano-type resonant spectra indicate destructive and instructive interferences between toroidal and electric dipoles, respectively. (b) Destructively induced toroidal mode at the 945-nm wavelength, with out-of-phase magnetic circulating distribution against that of the dipolar emitting source (Δy=0nm). (c) Constructively induced toroidal mode at the 1050-nm wavelength, with in-phase magnetic circulating distribution to the dipolar source (Δy=0nm). (d) The scattered powers in terms of multipoles when Δy=0nm.

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When the dipole emitter deviates from the central position of toroidal flat rings along radial directions, the interference interaction between toroidal and electric dipoles will lead to a different emission behavior. Without loss of generality, we consider a radial displacement along the x axis, for example, Δx=60nm (Fig. 3 ). It is found that the resonant toroidal response remains in the spectrum as well, indicating that the response of the toroidal dipole is inherent to the flat-ring geometry itself. On the other hand, although the radiating peak at 1050 nm under constructive interference (recorded by probe A/C) is not sensitive to the radial position of a dipole emitter, the emerging peak at 945 nm exhibits a dramatic emission enhancement (recorded by probe B) in contrast to the otherwise nonradiating dip. This can be regarded as a super-radiating effect with an emission power ever 10 times stronger than the 1050-nm enhanced emission.

 

Fig. 3 Far-field radiating power for the radial displacement Δx=60nm. The reference data are recorded by probe A/C for Δx=0nm.

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It was claimed that the resonant nanostructure is capable of modulating the emission direction of an adjacent emitter, which is promising for optical nanoantennas for purposes such as steering and/or collecting lights [25–30 ]. Although the dipolar emitter in the center of toroidal flat-ring structure radiates light in a donut-like pattern [Fig. 4(a) ], its emission mediated by the flat rings will evolve towards preferred directions so long as the emitter is shifted along radial directions. Figure 4(b) shows a moderate modification for the donut-like radiation pattern at 1050 nm. However, a pronounced directional radiation pattern can be obtained at the super-radiating wavelength 945 nm for Δx=60nm, exhibiting a property as is an optical nanoantenna [Fig. 4(c)]. From corresponding two-dimensional polar plots shown in Figs. 4(d)-4(f), the angular width of main-lobe radiation (angle between half-power radiation directions) can be greatly narrowed to an angle about 70 degree. In literatures, this directivity characteristic was explained by the versatile interference behaviors of the emitter coupling with the resonant mode in terms of multipoles, which will consequently determine the radiation directivity, intensity, as well as polarization [25–30 ].

 

Fig. 4 (a) Donut-like radiation pattern at 1050-nm wavelength for Δx=0nm. (b) Modulated radiation pattern at 1050-nm radiative wavelength for Δx=60nm. (c) Directional radiation pattern at 945-nm radiative wavelength for Δx=60nm. (d)-(f) Two-dimensional radiation in dependence of the polar angle θ under an azimuthal angle ϕ=90 degree, corresponding to (a)-(c), respectively. Note the 1-meter reference distance is used to set the radius of the virtual sphere for calculating field values of radiation pattern in Fig. 4.

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Figure 5 shows probed emission powers in dependence of radial displacement Δx. Due to different coupling efficiencies between the dipole emitter and the metastructure, emission powers can be dramatically modulated. When the emitter is in an off-axis position Δx=r1=100nm, the toroidal-dipole-mediated directional emission, |ExY|2, can reach a super-radiating maximum, roughly three orders stronger than radiation power of a bare emitter.

 

Fig. 5 Far-field radiation powers monitored by probe A (a), probe B (b), and probe C (c), under different radial displacements Δx.

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4. Summary

In summary, we explore the radiation characteristic of a toroidal-response-mediated dipole emitter under different coupling strategies with axial and radial (off-axis) displacements. Nonradiating dipole-dipole interaction is demonstrated by this coupling system of toroidal flat-rings and dipolar emitter, attributed to the destructive interference between overlapped radiative dipolar fields. Moreover, the otherwise nonradiating system will become a directionally super-radiating one, like an optical nanoantenna, so long as the dipolar emitter has a radial displacement. Our results are promising in manipulating the radiation power and direction of a single emitter, such as fluorescent molecule/atom and quantum dot, by utilizing the intriguing toroidal dipolar response based on the proposed flat-ring metastructure.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Nos. 11174051, 11374049, and 11204139), Natural Science Foundation of Jiangsu Province of China (BK20131283), and partially by the Jiangsu key laboratory for advanced metallic materials (BM2007204). J. L. was supported by the Jiangsu Innovation Projects for Graduate Student (KYLX15_0214).

References and links

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

2. 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]  

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

4. 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(44 Pt 2B), 046609 (2002). [CrossRef]   [PubMed]  

5. 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]  

6. 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]  

7. V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

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

9. 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]  

10. 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]  

11. 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]  

12. 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]  

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

14. V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014). [CrossRef]  

15. G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995). [CrossRef]  

16. 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]  

17. W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

18. 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]  

19. 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]  

20. 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]  

21. J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014). [CrossRef]  

22. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]  

23. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010). [CrossRef]   [PubMed]  

24. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

25. I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014). [CrossRef]   [PubMed]  

26. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008). [CrossRef]  

27. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010). [CrossRef]   [PubMed]  

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

29. W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22(13), 16178–16187 (2014). [CrossRef]   [PubMed]  

30. W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014). [CrossRef]  

References

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  1. L. B. Zel’dovich, “The relation between decay asymmetry and dipole moment of elementary particles,” Sov. Phys. JETP 6, 1148 (1958).
  2. 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]
  3. C. Schwartz, “Theory of hyperfine structure,” Phys. Rev. 97(2), 380–395 (1955).
    [Crossref]
  4. 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(44 Pt 2B), 046609 (2002).
    [Crossref] [PubMed]
  5. 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]
  6. 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]
  7. V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.
  8. K. Marinov, A. D. Boardman, V. A. Fedotov, and N. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007).
    [Crossref]
  9. 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]
  10. 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]
  11. 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]
  12. 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]
  13. Y. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
    [Crossref] [PubMed]
  14. V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
    [Crossref]
  15. G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995).
    [Crossref]
  16. 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]
  17. W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).
  18. 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]
  19. 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]
  20. 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]
  21. J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
    [Crossref]
  22. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
    [Crossref]
  23. Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
    [Crossref] [PubMed]
  24. A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).
  25. I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
    [Crossref] [PubMed]
  26. T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
    [Crossref]
  27. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
    [Crossref] [PubMed]
  28. W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core-shell nanoparticles,” ACS Nano 6(6), 5489–5497 (2012).
    [Crossref] [PubMed]
  29. W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22(13), 16178–16187 (2014).
    [Crossref] [PubMed]
  30. W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
    [Crossref]

2015 (5)

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. 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]

W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

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]

2014 (5)

V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
[Crossref]

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22(13), 16178–16187 (2014).
[Crossref] [PubMed]

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
[Crossref]

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]

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]

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]

2012 (4)

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]

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]

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

2010 (3)

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[Crossref] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (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]

2008 (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

2007 (1)

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

2005 (1)

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]

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(44 Pt 2B), 046609 (2002).
[Crossref] [PubMed]

1995 (1)

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995).
[Crossref]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[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]

Afanasiev, G. N.

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995).
[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]

Bartal, G.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[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]

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]

Brener, I.

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

Burckel, D. B.

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

Castro-López, M.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

Chau, Y. F.

Chen, Q.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

Chen, W. T.

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

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

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]

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

Curto, A. G.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

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]

Dong, Z.-G.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

Economou, E. N.

A. A. Basharin, M. Kafesaki, E. N. Economou, C. M. Soukoulis, V. A. Fedotov, V. Savinov, and N. I. Zheludev, “Dielectric metamaterials with toroidal dipolar response,” Phys. Rev. X 5(1), 011036 (2015).
[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. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
[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]

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]

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]

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

Hancu, I. M.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

Ho, Y. Z.

Hu, H.

Huang, Y. W.

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

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[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]

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
[Crossref]

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

Kreuzer, M. P.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

Kuttge, M.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (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]

Lei, B.

Li, J.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

Li, J. Q.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

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]

Liu, W.

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]

W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
[Crossref]

W. Liu, J. Zhang, B. Lei, H. Ma, W. Xie, and H. Hu, “Ultra-directional forward scattering by individual core-shell nanoparticles,” Opt. Express 22(13), 16178–16187 (2014).
[Crossref] [PubMed]

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

Liu, Y.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[Crossref] [PubMed]

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]

Ma, H.

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.

W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

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]

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
[Crossref]

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

Neshev, D. N.

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

Ni, P.

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]

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[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(44 Pt 2B), 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]

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. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
[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]

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

Schwartz, C.

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

Segerink, F. B.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Shao, J.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

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]

Stefani, F. D.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Stepanovsky, Y. P.

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995).
[Crossref]

Taminiau, T. H.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Tsai, D. P.

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

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

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

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(44 Pt 2B), 046609 (2002).
[Crossref] [PubMed]

van Hulst, N. F.

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

Vanhulst, N. F.

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

Volpe, G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[Crossref] [PubMed]

Wu, P. C.

Xie, W.

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, X. Q.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[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]

Zel’dovich, L. B.

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

Zentgraf, T.

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[Crossref] [PubMed]

Zhai, Y.

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

Zhang, J.

Zhang, J. F.

W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

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, 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]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[Crossref] [PubMed]

Zheludev, N.

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

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

V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
[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]

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

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

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]

ACS Nano (1)

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

Appl. Phys. Lett. (1)

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]

Chin. Phys. B (1)

W. Liu, A. E. Miroshnichenko, and Y. S. Kivshar, “Control of light scattering by nanoparticles with optically-induced magnetic responses,” Chin. Phys. B 23(4), 047806 (2014).
[Crossref]

J. Phys. A (1)

G. N. Afanasiev and Y. P. Stepanovsky, “The electromagnetic-field of elementary time-dependent toroidal sources,” J. Phys. A 28(16), 4565–4580 (1995).
[Crossref]

Laser Photonics Rev. (1)

W. Liu, J. F. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core-shell nanoparticles,” Laser Photonics Rev. 9(5), 564-570 (2015).

Nano Lett. (2)

I. M. Hancu, A. G. Curto, M. Castro-López, M. Kuttge, and N. F. van Hulst, “Multipolar interference for directed light emission,” Nano Lett. 14(1), 166–171 (2014).
[Crossref] [PubMed]

Y. Liu, T. Zentgraf, G. Bartal, and X. Zhang, “Transformational plasmon optics,” Nano Lett. 10(6), 1991–1997 (2010).
[Crossref] [PubMed]

Nat. Commun. (1)

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. Photonics (1)

T. H. Taminiau, F. D. Stefani, F. B. Segerink, and N. F. Vanhulst, “Optical antennas direct single-molecule emission,” Nat. Photonics 2(4), 234–237 (2008).
[Crossref]

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 (3)

Opt. Lett. (1)

Phys. Lett. A (1)

J. Li, J. Shao, J. Q. Li, X. Q. Yu, Z.-G. Dong, Q. Chen, and Y. Zhai, “Optical responses of magnetic-vortex resonance in double-disk metamaterial variations,” Phys. Lett. A 378(26-27), 1871–1875 (2014).
[Crossref]

Phys. Rev. (1)

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

Phys. Rev. B (3)

V. Savinov, V. A. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014).
[Crossref]

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972).
[Crossref]

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]

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(44 Pt 2B), 046609 (2002).
[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. Bao, X. Zhu, and Z. Fang, “Plasmonic toroidal dipolar response under radially polarized excitation,” Sci. Rep. 5, 11793 (2015).
[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. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal lasing spaser,” Sci. Rep. 3, 1237 (2013).
[Crossref] [PubMed]

Science (2)

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. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329(5994), 930–933 (2010).
[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).

Other (2)

V. Savinov, V. A. Fedotov, W. T. Chen, Y. W. Huang, D. P. Tsai, D. B. Burckel, I. Brener, and N. I. Zheludev, “Toroidal photonic metamaterial,” in Conference on Lasers and Electro-Optics (IEEE, 2012), pp. 1–2.

A. Taflove and S. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2000).

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Figures (5)

Fig. 1
Fig. 1 Schematic of the metastructure composed of double metallic flat-rings with a silicon-oxide gap spacer. Note that the origin “O” of the coordinate system is located at the center of metastructure.
Fig. 2
Fig. 2 (a) Far-field radiating power for different axial displacements Δ y . The nonradiating dips and super-radiative peaks in the Fano-type resonant spectra indicate destructive and instructive interferences between toroidal and electric dipoles, respectively. (b) Destructively induced toroidal mode at the 945-nm wavelength, with out-of-phase magnetic circulating distribution against that of the dipolar emitting source ( Δ y = 0 nm ). (c) Constructively induced toroidal mode at the 1050-nm wavelength, with in-phase magnetic circulating distribution to the dipolar source ( Δ y = 0 nm ). (d) The scattered powers in terms of multipoles when Δ y = 0 nm .
Fig. 3
Fig. 3 Far-field radiating power for the radial displacement Δ x = 60 nm . The reference data are recorded by probe A/C for Δ x = 0 nm .
Fig. 4
Fig. 4 (a) Donut-like radiation pattern at 1050-nm wavelength for Δ x = 0 nm . (b) Modulated radiation pattern at 1050-nm radiative wavelength for Δ x = 60 nm . (c) Directional radiation pattern at 945-nm radiative wavelength for Δ x = 60 nm . (d)-(f) Two-dimensional radiation in dependence of the polar angle θ under an azimuthal angle ϕ = 90 degree, corresponding to (a)-(c), respectively. Note the 1-meter reference distance is used to set the radius of the virtual sphere for calculating field values of radiation pattern in Fig. 4.
Fig. 5
Fig. 5 Far-field radiation powers monitored by probe A (a), probe B (b), and probe C (c), under different radial displacements Δ x .

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