- Email: [email protected]
Tailoring the excitation of two kinds of toroidal dipoles in all-dielectric metasurfaces
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Tailoring the excitation of two kinds of toroidal dipoles in alldielectric metasurfaces
T
Xiangjun Lia,b,c,*, Jie Yina,b, Zihao Liua,b, Yi Wangb, Zhi Honga a b c
Centre for THz Research, China Jiliang University, Hangzhou, China College of Information Engineering, China Jiliang University, Hangzhou, China Key Laboratory of Electromagnetic Wave Information Technology and Metrology of Zhejiang Province, China Jiliang University, Hangzhou, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Toroidal dipole All-dielectric metasurface High-quality factor
In this paper, the simultaneous excitation of the toroidal dipoles within the unit cell and between neighboring unit cells is theoretically proposed and demonstrated in an all-dielectric metasurface composed of silicon bars. We referred to the toroidal dipole moment within the unit cell as the intra-toroidal dipole moment, and the toroidal dipole moment between the unit cells as the intertoroidal dipole moment. We found that the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can be dramatically increased by decreasing the distance between the adjacent silicon bars, so the electric and magnetic field is significantly confined both inside and between the silicon bars. Hence the radiation loss is greatly suppressed and a toroidal resonance with the high Q-factor of 63,000 is achieved. The proposed all-dielectric metasurface is low loss, simple structure and operating in the near infrared regime, which has potential applications in biochemical sensing, non-linear devices, slow light-based devices, and other optical devices.
1. Introduction Toroidal dipole, a crucial part of multipoles decomposition, was neglected for a long time since Yakov Zel'dovich found that there must be a toroidal dipole moment to rationalize the weak interaction violating parity symmetry in 1958 [1]. The toroidal dipole is determined by electrical currents circulating on a surface of a torus along its meridians. The current can be regarded as a series of the head-to-tail magnetic dipoles circulating along closed loops [2,3]. It is revealed that toroidal dipole can exist in natural media such as ferroelectric systems [4,5], macromolecules [6] and glasses [7]. However, such natural toroidal response is hardly observable because their feeble contribution comparing of electric and magnetic dipoles. The situation has been changed since the tremendous degree of freedom offered by metasurfaces enabled the toroidal dipole to become the main component in multipoles resonances. In 2010, toroidal dipole was experimentally demonstrated in the microwave regime with a periodical array of four metallic split-ring resonators [8]. Subsequently, the excitation of toroidal dipole has been widely explored at microwave [9–13], THz [14–18], infrared [19,20], and visible spectral ranges [21–23]. Besides, due to the identical far field radiation mode of the toroidal dipole and the electric dipole, it has been widely reported that the tailoring the excitation of toroidal dipole to achieve an anapole mode [24–27], where the electrical dipole and toroidal dipole form destructive interference between each other resulting in complete non-radiative mode. So far, the previous works have intensely studied on the excitation of toroidal dipole moment within the unit cell, where
⁎
Corresponding author at: Centre for THz Research, China Jiliang University, Hangzhou, China. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.ijleo.2019.163502 Received 16 June 2019; Accepted 30 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
clockwise-anticlockwise displacement current loops formed in a unit cell [8–23]. Nevertheless, the excitation of toroidal dipole moment between the neighboring unit cells was rarely reported, which is produced by reverse current loops that belong to different nanoparticle. In 2017, Gu et al. designed the double split ring resonators on an ultrathin self-supporting gold membrane [28]. They demonstrated the excitation of two kinds of toroidal dipole modes in the metamaterials. One is toroidal dipole within the unit cell, other is toroidal dipole between unit cell. Especially, the two toroidal dipole modes appear at different resonance wavelengths. In 2018, Su Xu et al. also studied and experimental verified the excitation of the two distinct types of the toroidal dipole modes in metasurfaces composed of trimer clusters of high-index dielectric particles in microwave region [29]. Similarly, the two distinct toroidal dipole modes also appear at different resonance wavelengths. As we know, the toroidal dipole moment can be excited within the unit cell [8–23], thus, it must be excited simultaneously between the neighboring unit cells when the distance between adjacent particles is small in periodic metasurfaces. In other words, the two distinct types of the toroidal dipole modes should be able to be excited simultaneously at the same resonant wavelength. However, the view has not been studied so far, because of the weak coupling between unit cells. In this paper, we theoretically proposed and verified the simultaneous excitation of two kinds of the toroidal dipoles in an alldielectric metasurface consisting of silicon bar arrays. We referred to the toroidal dipole moment within the unit cell as the intratoroidal moment and the toroidal dipole moment between the unit cells as the inter-toroidal moment. We found the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can increases quickly by decreasing the distance between the adjacent silicon bars, hence the radiation loss is greatly suppressed and toroidal resonance with the high Q-factor of 63,000 is achieved. Such all-dielectric metasurface may be applied in light-matter interactions, such as biochemical sensing, nonlinear devices, slow light-based devices, and other optical devices. 2. Design and simulation results 2.1. Multipole decomposition In order to investigate the electromagnetic and scattering properties of the metasurfaces, Cartesian scattered powers for multipole moments are calculated based on the induced current inside of the nanoparticle [30–32]. In this way, the multipole contributions can be identified. In our calculations, the dipole moments of the nanoparticle for the electric, magnetic, toroidal, contributions read as
Pα =
1 iω
1 2c
Mα = Tα =
∫ Jα d3r
1 10c
(1)
∫ (r × J )α d3r
(2)
∫ [(r⋅J ) rα + 2r 2Jα]d3r
(3)
where J is the induced volume current density, r is position vector, c is the speed of light in vacuum and α = x,y,z. The electric and magnetic quadrupole moments are expressed as
QEαβ =
1 2iω
∫ [(rα Jβ + rβ Jα ) − 23 (r⋅J ) δαβ ]d3r
(4)
QMαβ =
1 3c
∫ [(r × J )α rβ + (r × J )β rα]d3r
(5)
δαβ where is delta function and α,β = x,y,z. The scattering powers associated with each multipole moment are given by IPα =
2ω4 |Pα |2 3c3 2ω4 |Mα |2 3c3
IMα = ITα =
(6)
2ω6 3c5
(7)
|Tα |2
IQE =
ω6 5c5
IQM =
ω6 20c5
(8)
∑ |QEαβ |2
(9)
∑ |QMαβ |2
(10)
2.2. All-dielectric metasurfaces The schematic of the designed all-dielectric silicon bar arrays (refractive index n = 3.45, lossless) is shown in the Fig. 1. The silicon bar’s geometrical parameters are a = 700 nm, b =300 nm and h =200 nm. The surrounding medium is air (n = 1), and the 2
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 1. Schematic of the designed all-dielectric silicon bar arrays. The metasurface is illustrated by a normal incident wave with the electric polarization along the x axis. a =600 nm, b =300 nm and h =200 nm.
lattice periods are fixed along the x and y directions with Px and Py, respectively. Numerical calculations are conducted by using Finite difference time domain (FDTD) simulation software. In FDTD simulations, the metasurface is illuminated by a normally incident plane wave, which polarized along vertical bars. Periodic boundary conditions are used in both x and y directions, and perfectly matched layers are applied in z-direction. As a first remark, when Px =750 nm and Py =1000 nm, the simulated transmission spectrum in the wavelength range of 1000 nm to 1500 nm is given in Fig. 2(a). We can notice that there are two strong and clear resonant dips at 1115 nm and 1330 nm, respectively. The Q-factor of the resonance at 1330 nm is manifest larger than that of the resonance at 1115 nm. In order to demonstrate the role of the dipole excitation at the two resonances, Cartesian scattered powers for multipole moments are calculated based on the induced current inside of the silicon bar in the metasurface, as given in Fig. 2(b), where Px, My, Tx, QE and QM are the electric dipole in x-direction, magnetic dipole in y-direction, toroidal dipole in x-direction, electric quadrupole and magnetic quadrupole, respectively. Here, higher order dipoles are ignored because of the extremely smaller intensity on the scattered powers. We can see that the scattering power of magnetic dipole is the largest at 1115 nm, which proves it is a magnetic dipole resonance (MR). For the resonance at 1330 nm, the toroidal dipole is the dominating multipole contributions of scattering powers. It is approximately 2 times stronger than the electric dipole Px, about 7 times larger than the magnetic quadrupole QM and around 2 orders stronger than the electric quadrupole QE and the magnetic dipole My. It is noteworthy that the resonance at 1330 nm is an asymmetric dip, which may originate from the interference of the resonance of toroidal dipolar nature at 1330 nm with a broad sub-radiant, high-transmission background [33]. For the sake of simplicity, we call the resonance at 1330 nm as toroidal dipole resonance (TR). Here, the Q value of TR is calculated by fitting the transmission spectrum of Fig. 2(a) with the following Fano formula [34,35],
Fig. 2. (a) Transmission spectrum of the metasurface in the wavelength range of 1000 nm–1500 nm. (b) Five scattering powers of multipoles decomposition for the investigated metasurface, where Px, My, Tx, QE and QM are the electric dipole in x-direction, magnetic dipole in y-direction, toroidal dipole in x-direction, electric quadrupole and magnetic quadrupole, respectively. The log scale of y-axis is chosen in order to display more clearly the contribution results of the multipole decomposition. The vertical gray line labels the resonance wavelengths. The lattice periods are Px =750 nm and Py =1000 nm. 3
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 3. (a) E-field and (b) M-field enhancement profiles at the toroidal dipole resonance wavelength of 1330 nm when Px =750 nm and Py =1000 nm. The black arrows represent the directions of electric or magnetic fields.
I∝
(Fγ + ω − ω0 )2 (ω − ω0 )2 + γ 2
(11)
where F is the Fano parameter, γ and ω0 represent the width and position of the resonance. The Q value is calculated by ω0 /γ. For the case, the fitting parameter are F = 1, ω0 = 1418 rad/s, γ = 4.05 rad/s, respectively. The Q factor of TR is about 350. Fig. 3 shows the relative enhancements of the electric and magnetic nearfields at the x–y and y–z planes of the silicon bar, respectively. The black arrows represent the field vector distribution. The field enhancements are calculated at the wavelengths corresponding to the toroidal dipole resonance for Px =750 nm and Py =1000 nm. It is found that there are opposite circular displacement currents on two sides of the silicon bar simultaneously, each of them could generate a magnetic field, which upside is along +z axis, and -z axis for downside. These induced magnetic field patterns with low radiation into far filed can produce a circular magnetic moment confined in the volume of the silicon bar. Therefore, the head-to-tail magnetic moment connected leads to the formation of toroidal dipole moment along the x-axis [8,36]. In our metasurface, both electric and magnetic fields are confined inside the silicon bars (see Fig. 3), so each silicon bar can be seen as an effective waveguide [37–40], the intercoupling between the waveguides may play an essential role in the Q value of the TR. We firstly adjust the value of the lattice period in y-direction from 1000 nm to 720 nm. Fig. 4 shows the transmission spectra of the metasurface at four different lattice periods. We can see that both wavelength and bandwidth of MR remains mostly unaltered, but the wavelength of the TR slightly shifts from 1330 nm to 1308 nm. Meanwhile, the bandwidth of the TR has a dramatic decrease. To further clearly show the characters of Q value at TR, the Q value as a function of the Py for the all-dielectric metasurface is shown in Fig. 5. Here, Q value is also calculated by ω0 /γ. It is revealed that the Q value increases slowly, as the lattice period Py alters from 1000 nm to 850 nm, but rises quickly when Py changes from 850 nm to 720 nm. We can notice that the Q value reaches 63,000 when Py =720 nm, which is nearly 180 times stronger than that of Py =1000 nm. To study the collective oscillation in the metasurface, the electric field (x–y plane), magnetic field (y–z plane) and illustration of the toroidal dipole moments in multiple unit cells is given in Fig. 6 (Px = Py =750 nm). We referred to the toroidal dipole moment
Fig. 4. Transmission spectra of the metasurface at different lattice periods Py, where Px is fixed of 750 nm. The inset shows the magnified spectra of the TR. 4
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 5. The quality-factor dependence of TR on the lattice period Py, ranging from 1000 nm to 720 nm.
Fig. 6. The electric field (x–y plane), magnetic field (y–z plane) and illustration of the toroidal dipole moments in multiple unit cells. j (green arrows) represent the displacement currents and m (yellow arrows) represent the magnetic moments. T1 and T2 (claret arrows) represent intratoroidal dipole moment and inter-toroidal dipole moment, respectively. Small black arrows represent the directions of electric or magnetic fields (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
within the unit cell as the intra-toroidal dipole moment and the toroidal dipole moment between the unit cells as the inter-toroidal dipole moment. As shown in Fig. 6, there are two kinds of toroidal dipole moments: one is intra-toroidal dipole moments (named as T1), and the other is the inter-toroidal dipole moment (named as T2). For the intra-toroidal dipole moment, there are opposite circular displacement currents on two sides of one silicon bar simultaneously (such as j1 and j2), each of them could generate a magnetic field. These induced magnetic field patterns can produce a circular magnetic moment (intra-magnetic moment, such as m1) confined in the volume of the silicon bar. Therefore, the head-to-tail magnetic moment connected leads to the formation of intra-toroidal dipole moment along the x-axis [8,36], which has been verified in Fig. 3. The inter-toroidal dipole moment is produced by circular displacement currents of neighboring silicon bars along y-axis (such as j2 and j3). Interestingly, the circular displacement currents and the corresponding magnetic moments in the lower part of the silicon bar (such as j3) and upper part of the neighboring one (such as j2) are opposite to each other. The head-to-tail configuration of magnetic moment (inter-magnetic moment, such as m2) results in strong intercoupling between the neighboring silicon bars and indicates the excitation of inter-toroidal dipole moment oriented along the x-axis. In order to analyze the contribution of inter- and intra-toroidal dipole moments when decreasing the distance between the adjacent silicon bars, the distributions of the magnetic field in the y–z plane at the TR are illustrated in Fig. 7(a) ∼ 7(d) when Py =1000 nm, 900 nm, 800 nm and 720 nm, respectively. The intensity of inter-magnetic moment and intra-magnetic moment can reflect the intensity of inter-toroidal dipole moment and intra-toroidal dipole moment, respectively. We notice that the intensity of both inter- and intra-magnetic moment can be dramatically increased by decreasing the distance between the adjacent silicon bars. This means the inter- and intra-toroidal dipole moments be significantly excited by changing the lattice period in y-direction. The electric and magnetic field of the incident wave is significantly confined both inside and between the silicon bars as the toroidal dipole moments increase, resulting in achievements of the high-Q resonance of the metasurface [8,9,22]. Interestingly, the Q value at TR as a function of the Py keeps increased monotonically as given in Fig. 5, which means the inter-toroidal dipole moment and intra5
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 7. (a), (b), (c), (d) Distributions of the magnetic field in the y–z plane at TR when Py =1000 nm, 900 nm, 800 nm and 720 nm, respectively. m1 is the intra-magnetic moment and m2 is the inter-magnetic moment.
one may not interfere with each other. We believe that the reason for this phenomenon may be that inter-toroidal dipole moment and intra-one produce in different locations, which is different from the excitation of anapole mode [24–27]. In the above works, the silicon bar arrays are supposed to embedded in air. However, the TR of our proposed metasurface can be also excited effectively with the presence of a substrate. In Fig. 8, the red solid line represents the transmission spectrum of the silicon bar arrays that are deposited on a semi-infinite quartz substrate (n = 1.46), and the black solid line represents the transmission spectrum of the silicon bars that is freestanding in air, in both cases, Px = Py =750 nm. Compared with the metasurface without substrate, we can see that the wavelength of the TR for the metasurface with a substrate exhibits a red-shift from 1310 nm to 1330 nm, and the Q value of the resonance decreases slightly from 6650 to 4140, because of the dielectric losses of the substrate. This means that the influence of a substrate on the Q factor of the toroidal resonance is much small. Besides, the Q-factor may be further enlarged by carefully designing the geometry parameters of silicon bars.
3. Conclusion In conclusion, we have presented and numerically investigated the simultaneous excitation of the intra-toroidal dipole moment
Fig. 8. Transmission spectra of silicon bar arrays deposited on a semi-infinite quartz substrate (red solid line) and disposed in air (black solid line) when Px = Py =750 nm, respectively. The inset shows the magnified spectrum of the TRs (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). 6
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
and inter-toroidal dipole moment in all-dielectric metasurface consisting of silicon bar arrays in the near-infrared spectral region. Cartesian multipolar decomposition indicates that a narrowband toroidal resonance is observed in the metasurface. We found that the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can be dramatically increased by decreasing the distance between the adjacent silicon bars. Thus, the electric and magnetic field of the incident wave is significantly confined both inside and between the silicon bars, resulting in achievements of high-Q toroidal resonance of the metasurface. Besides, we also demonstrated that the TR can still be effectively excited with a small decrease of Q value when the silicon metasurface is deposited on a quartz substrate. The proposed all-dielectric metasurface has the advantages of low loss and simple structure so that it may have wide applications in light-matter interactions, such as biochemical sensing, non-linear devices, slow light-based devices, and other optical devices. Acknowledgments The work was supported by the National Natural Science Foundation of China (Grant Nos. 61875179, 61875251 and 51704267) and the Natural Science Foundation of Zhejiang of China (Grant No. LY17E04002). References [1] I.B. Zel’Dovich, The relation between decay asymmetry and dipole moment of elementary particles, Sov. Phys. JETP 6 (1958) 1148. [2] V.M. Dubovik, L.A. Tosunian, V.V. Tugushev, Axial toroidal moments in electrodynamics and solid-state physics, Zh. Eksp. Teor. Fiz. 90 (1986) 590–605. [3] G. Thorner, J. Kiat, C. Bogicevic, I. Kornev, Axial hypertoroidal moment in a ferroelectric nanotorus: a way to switch local polarization, Phys. Rev. B Condens. Matter Mater. Phys. 89 (2014) 220103, , https://doi.org/10.1103/PhysRevB.89.220103. [4] I.I. Naumov, L. Bellaiche, H.X. Fu, Unusual phase transitions in ferroelectric nanodisks and nanorods, Nature 432 (2004) 737–740, https://doi.org/10.1038/ nature03107. [5] N.A. Spaldin, M. Fiebig, M. Mostovoy, The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect, J. Phys. Condens. Matter 20 (2008) 434203, , https://doi.org/10.1088/0953-8984/20/43/434203. [6] A. Ceulemans, L.F. Chibotaru, P.W. Fowler, Molecular anapole moments, Phys. Rev. Lett. 80 (1998) 1861–1864, https://doi.org/10.1103/PhysRevLett.80.1861. [7] Y. Yamaguchi, T. Kimura, Magnetoelectric control of frozen state in a toroidal glass, Nat. Commun. 4 (2013) 2063, https://doi.org/10.1038/ncomms3063. [8] T. Kaelberer, V.A. Fedotov, N. Papasimakis, D.P. Tsai, N.I. Zheludev, Toroidal dipolar response in a metamaterial, Science 330 (2010) 1510–1512, https://doi. org/10.1126/science.1197172. [9] Z. Dong, P. Ni, J. Zhu, X. Yin, X. Zhang, Toroidal dipole response in a multifold double-ring metamaterial, Opt. Express 20 (2012) 13065–13070, https://doi.org/ 10.1364/OE.20.013065. [10] L.Y. Guo, M.H. Li, Q.W. Ye, B.X. Xiao, H.L. Yang, Electric toroidal dipole response in split-ring resonator metamaterials, Eur. Phys. J. B. 85 (2012) 1–5, https:// doi.org/10.1140/epjb/e2012-20935-3. [11] Y. Fan, Z. Wei, H. Li, H. Chen, C.M. Soukoulis, Low-loss and high-Q planar metamaterial with toroidal moment, Phys. Rev. B 87 (2013) 115417, , https://doi.org/ 10.1103/PhysRevB.87.115417. [12] T.A. Raybould, V.A. Fedotov, N. Papasimakis, I. Kuprov, I.J. Youngs, W.T. Chen, D.P. Tsai, N.I. Zheludev, Toroidal circular dichroism, Phys. Rev. B 94 (2016), https://doi.org/10.1103/PhysRevB.94.035119. [13] S. Han, L. Cong, F. Gao, R. Singh, H. Yang, Observation of Fano resonance and classical analog of electromagnetically induced transparency in toroidal metamaterials, Ann. Phys.-Berlin 528 (2016) 352–357, https://doi.org/10.1002/andp.201600016. [14] L. Cong, Y.K. Srivastava, R. Singh, Tailoring the multipoles in THz toroidal metamaterials, Appl. Phys. Lett. 111 (2017) 081108, , https://doi.org/10.1063/1. 4993670. [15] Y. Fan, F. Zhang, N. Shen, Q. Fu, Z. Wei, H. Li, C.M. Soukoulis, Achieving a high-Q response in metamaterials by manipulating the toroidal excitations, Phys. Rev. A 97 (2018) 033816, , https://doi.org/10.1103/PhysRevA.97.033816. [16] M. Gupta, V. Savinov, N. Xu, L. Cong, G. Dayal, S. Wang, W. Zhang, N.I. Zheludev, R. Singh, Sharp toroidal resonances in planar terahertz metasurfaces, Adv. Mater. 28 (2016) 8206–8211, https://doi.org/10.1002/adma.201601611. [17] M. Gupta, Y.K. Srivastava, R. Singh, A toroidal metamaterial switch, Adv. Mater. 30 (2018), https://doi.org/10.1002/adma.201704845. [18] M. Gupta, Y.K. Srivastava, M. Manjappa, R. Singh, Sensing with toroidal metamaterial, Appl. Phys. Lett. 110 (2017) 121108, , https://doi.org/10.1063/1. 4978672. [19] B. Han, X. Li, C. Sui, J. Diao, X. Jing, Z. Hong, Analog of electromagnetically induced transparency in an E-shaped all-dielectric metasurface based on toroidal dipolar response, Opt. Mater. Express 8 (2018) 2197–2207, https://doi.org/10.1364/OME.8.002197. [20] Z. Liu, S. Du, A. Cui, Z. Li, Y. Fan, S. Chen, W. Li, J. Li, C. Gu, High-Quality-Factor Mid-Infrared toroidal excitation in folded 3D metamaterials, Adv. Mater. 29 (2017) 1606298, , https://doi.org/10.1002/adma.201606298. [21] B. Oeguet, N. Talebi, R. Vogelgesang, W. Sigle, P.A. van Aken, Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible, Nano Lett. 12 (2012) 5239–5244, https://doi.org/10.1021/nl302418n. [22] Y.W. Huang, W.T. Chen, P.C. Wu, V. Fedotov, V. Savinov, Y.Z. Ho, Y.F. Chau, N.I. Zheludev, D.P. Tsai, Design of plasmonic toroidal metamaterials at optical frequencies, Opt. Express 20 (2012) 1760–1768, https://doi.org/10.1364/OE.20.001760. [23] F. Huang, Z. Wang, X. Lu, J. Zhang, K. Min, W. Lin, R. Ti, T. Xu, J. He, C. Yue, J. Zhu, Peculiar magnetism of BiFeO3 nanoparticles with size approaching the period of the spiral spin structure, Sci. Rep. 3 (2013) 2907, https://doi.org/10.1038/srep02907. [24] P.C. Wu, C.Y. Liao, V. Savinov, T.L. Chung, W.T. Chen, Y.W. Huang, P.R. Wu, Y.H. Chen, A.Q. Liu, N.I. Zheludev, D.P. Tsai, Optical anapole metamaterial, ACS Nano 12 (2018) 1920–1927, https://doi.org/10.1021/acsnano.7b08828. [25] L. Wei, Z. Xi, N. Bhattacharya, H.P. Urbach, Excitation of the radiationless anapole mode, Optica 3 (2016) 799, https://doi.org/10.1364/OPTICA.3.000799. [26] S. Liu, Z. Wang, W. Wang, J. Chen, Z. Chen, High Q-factor with the excitation of anapole modes in dielectric split nanodisk arrays, Opt. Express 25 (2017) 22375–22387, https://doi.org/10.1364/OE.25.022375. [27] J.F. Algorri, D.C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, J.M. Sanchez-Pena, Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids, Opt. Express 27 (2019) 6320–6330, https://doi.org/10.1364/OE.27.006320. [28] S. Yang, Z. Liu, L. Jin, W. Li, S. Zhang, J. Li, C. Gu, Surface plasmon polariton mediated multiple toroidal resonances in 3D folding metamaterials, ACS Photonics 4 (2017) 2650–2658, https://doi.org/10.1021/acsphotonics.7b00529. [29] S. Xu, A. Sayanskiy, A.S. Kupriianov, V.R. Tuz, P. Kapitanova, H. Sun, W. Han, Y.S. Kivshar, Experimental observation of toroidal dipole modes in All-Dielectric metasurfaces, Adv. Opt. Mater. 7 (2019) 1801166, , https://doi.org/10.1002/adom.201801166. [30] A.A. Basharin, M. Kafesaki, E.N. Economou, C.M. Soukoulis, V.A. Fedotov, V. Savinov, N.I. Zheludev, Dielectric metamaterials with toroidal dipolar response, Phys. Rev. X 5 (2015) 011036, , https://doi.org/10.1103/PhysRevX.5.011036. [31] E.E. Radescu, G. Vaman, Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles, Phys. Rev. E 65 (2002) 046609, , https://doi.org/10.1103/PhysRevE.65.046609. [32] V. Savinov, V.A. Fedotov, N.I. Zheludev, Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials, Phys. Rev. B 89 (2014)
7
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
025112, , https://doi.org/10.1103/PhysRevB.89.205112. [33] N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Soennichsen, H. Giessen, Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing, Nano Lett. 10 (2010) 1103–1107, https://doi.org/10.1021/nl902621d. [34] S.H. Fan, Sharp asymmetric line shapes in side-coupled waveguide-cavity systems, Appl. Phys. Lett. 80 (2002) 908–910, https://doi.org/10.1063/1.1448174. [35] B. Luk’Yanchuk, N.I. Zheludev, S.A. Maier, N.J. Halas, P. Nordlander, H. Giessen, C.T. Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat. Mater. 9 (2010) 707–715, https://doi.org/10.1038/NMAT2810. [36] K. Marinov, A.D. Boardman, V.A. Fedotov, N. Zheludev, Toroidal metamaterial, New J. Phys. 9 (2007), https://doi.org/10.1088/1367-2630/9/9/324. [37] M. Khorasaninejad, F. Capasso, Broadband multifunctional efficient Meta-Gratings based on dielectric waveguide phase shifters, Nano Lett. 15 (2015) 6709–6715, https://doi.org/10.1021/acs.nanolett.5b02524. [38] A. Arbabi, Y. Horie, A.J. Ball, M. Bagheri, A. Faraon, Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays, Nat. Commun. 6 (2015) 7069, https://doi.org/10.1038/ncomms8069. [39] M. Khorasaninejad, A.Y. Zhu, C. Roques-Carmes, W.T. Chen, J. Oh, I. Mishra, R.C. Devlin, F. Capasso, Polarization-Insensitive metalenses at visible wavelengths, Nano Lett. 16 (2016) 7229–7234, https://doi.org/10.1021/acs.nanolett.6b03626. [40] C. Sui, X. Li, T. Lang, X. Jing, J. Liu, Z. Hong, High Q-Factor resonance in a symmetric array of All-Dielectric bars, Appl. Sci. 8 (2018) 161, https://doi.org/10. 3390/app8020161.
8
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.com/locate/ijleo
Original research article
Tailoring the excitation of two kinds of toroidal dipoles in alldielectric metasurfaces
T
Xiangjun Lia,b,c,*, Jie Yina,b, Zihao Liua,b, Yi Wangb, Zhi Honga a b c
Centre for THz Research, China Jiliang University, Hangzhou, China College of Information Engineering, China Jiliang University, Hangzhou, China Key Laboratory of Electromagnetic Wave Information Technology and Metrology of Zhejiang Province, China Jiliang University, Hangzhou, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Toroidal dipole All-dielectric metasurface High-quality factor
In this paper, the simultaneous excitation of the toroidal dipoles within the unit cell and between neighboring unit cells is theoretically proposed and demonstrated in an all-dielectric metasurface composed of silicon bars. We referred to the toroidal dipole moment within the unit cell as the intra-toroidal dipole moment, and the toroidal dipole moment between the unit cells as the intertoroidal dipole moment. We found that the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can be dramatically increased by decreasing the distance between the adjacent silicon bars, so the electric and magnetic field is significantly confined both inside and between the silicon bars. Hence the radiation loss is greatly suppressed and a toroidal resonance with the high Q-factor of 63,000 is achieved. The proposed all-dielectric metasurface is low loss, simple structure and operating in the near infrared regime, which has potential applications in biochemical sensing, non-linear devices, slow light-based devices, and other optical devices.
1. Introduction Toroidal dipole, a crucial part of multipoles decomposition, was neglected for a long time since Yakov Zel'dovich found that there must be a toroidal dipole moment to rationalize the weak interaction violating parity symmetry in 1958 [1]. The toroidal dipole is determined by electrical currents circulating on a surface of a torus along its meridians. The current can be regarded as a series of the head-to-tail magnetic dipoles circulating along closed loops [2,3]. It is revealed that toroidal dipole can exist in natural media such as ferroelectric systems [4,5], macromolecules [6] and glasses [7]. However, such natural toroidal response is hardly observable because their feeble contribution comparing of electric and magnetic dipoles. The situation has been changed since the tremendous degree of freedom offered by metasurfaces enabled the toroidal dipole to become the main component in multipoles resonances. In 2010, toroidal dipole was experimentally demonstrated in the microwave regime with a periodical array of four metallic split-ring resonators [8]. Subsequently, the excitation of toroidal dipole has been widely explored at microwave [9–13], THz [14–18], infrared [19,20], and visible spectral ranges [21–23]. Besides, due to the identical far field radiation mode of the toroidal dipole and the electric dipole, it has been widely reported that the tailoring the excitation of toroidal dipole to achieve an anapole mode [24–27], where the electrical dipole and toroidal dipole form destructive interference between each other resulting in complete non-radiative mode. So far, the previous works have intensely studied on the excitation of toroidal dipole moment within the unit cell, where
⁎
Corresponding author at: Centre for THz Research, China Jiliang University, Hangzhou, China. E-mail address: [email protected] (X. Li).
https://doi.org/10.1016/j.ijleo.2019.163502 Received 16 June 2019; Accepted 30 September 2019 0030-4026/ © 2019 Elsevier GmbH. All rights reserved.
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
clockwise-anticlockwise displacement current loops formed in a unit cell [8–23]. Nevertheless, the excitation of toroidal dipole moment between the neighboring unit cells was rarely reported, which is produced by reverse current loops that belong to different nanoparticle. In 2017, Gu et al. designed the double split ring resonators on an ultrathin self-supporting gold membrane [28]. They demonstrated the excitation of two kinds of toroidal dipole modes in the metamaterials. One is toroidal dipole within the unit cell, other is toroidal dipole between unit cell. Especially, the two toroidal dipole modes appear at different resonance wavelengths. In 2018, Su Xu et al. also studied and experimental verified the excitation of the two distinct types of the toroidal dipole modes in metasurfaces composed of trimer clusters of high-index dielectric particles in microwave region [29]. Similarly, the two distinct toroidal dipole modes also appear at different resonance wavelengths. As we know, the toroidal dipole moment can be excited within the unit cell [8–23], thus, it must be excited simultaneously between the neighboring unit cells when the distance between adjacent particles is small in periodic metasurfaces. In other words, the two distinct types of the toroidal dipole modes should be able to be excited simultaneously at the same resonant wavelength. However, the view has not been studied so far, because of the weak coupling between unit cells. In this paper, we theoretically proposed and verified the simultaneous excitation of two kinds of the toroidal dipoles in an alldielectric metasurface consisting of silicon bar arrays. We referred to the toroidal dipole moment within the unit cell as the intratoroidal moment and the toroidal dipole moment between the unit cells as the inter-toroidal moment. We found the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can increases quickly by decreasing the distance between the adjacent silicon bars, hence the radiation loss is greatly suppressed and toroidal resonance with the high Q-factor of 63,000 is achieved. Such all-dielectric metasurface may be applied in light-matter interactions, such as biochemical sensing, nonlinear devices, slow light-based devices, and other optical devices. 2. Design and simulation results 2.1. Multipole decomposition In order to investigate the electromagnetic and scattering properties of the metasurfaces, Cartesian scattered powers for multipole moments are calculated based on the induced current inside of the nanoparticle [30–32]. In this way, the multipole contributions can be identified. In our calculations, the dipole moments of the nanoparticle for the electric, magnetic, toroidal, contributions read as
Pα =
1 iω
1 2c
Mα = Tα =
∫ Jα d3r
1 10c
(1)
∫ (r × J )α d3r
(2)
∫ [(r⋅J ) rα + 2r 2Jα]d3r
(3)
where J is the induced volume current density, r is position vector, c is the speed of light in vacuum and α = x,y,z. The electric and magnetic quadrupole moments are expressed as
QEαβ =
1 2iω
∫ [(rα Jβ + rβ Jα ) − 23 (r⋅J ) δαβ ]d3r
(4)
QMαβ =
1 3c
∫ [(r × J )α rβ + (r × J )β rα]d3r
(5)
δαβ where is delta function and α,β = x,y,z. The scattering powers associated with each multipole moment are given by IPα =
2ω4 |Pα |2 3c3 2ω4 |Mα |2 3c3
IMα = ITα =
(6)
2ω6 3c5
(7)
|Tα |2
IQE =
ω6 5c5
IQM =
ω6 20c5
(8)
∑ |QEαβ |2
(9)
∑ |QMαβ |2
(10)
2.2. All-dielectric metasurfaces The schematic of the designed all-dielectric silicon bar arrays (refractive index n = 3.45, lossless) is shown in the Fig. 1. The silicon bar’s geometrical parameters are a = 700 nm, b =300 nm and h =200 nm. The surrounding medium is air (n = 1), and the 2
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 1. Schematic of the designed all-dielectric silicon bar arrays. The metasurface is illustrated by a normal incident wave with the electric polarization along the x axis. a =600 nm, b =300 nm and h =200 nm.
lattice periods are fixed along the x and y directions with Px and Py, respectively. Numerical calculations are conducted by using Finite difference time domain (FDTD) simulation software. In FDTD simulations, the metasurface is illuminated by a normally incident plane wave, which polarized along vertical bars. Periodic boundary conditions are used in both x and y directions, and perfectly matched layers are applied in z-direction. As a first remark, when Px =750 nm and Py =1000 nm, the simulated transmission spectrum in the wavelength range of 1000 nm to 1500 nm is given in Fig. 2(a). We can notice that there are two strong and clear resonant dips at 1115 nm and 1330 nm, respectively. The Q-factor of the resonance at 1330 nm is manifest larger than that of the resonance at 1115 nm. In order to demonstrate the role of the dipole excitation at the two resonances, Cartesian scattered powers for multipole moments are calculated based on the induced current inside of the silicon bar in the metasurface, as given in Fig. 2(b), where Px, My, Tx, QE and QM are the electric dipole in x-direction, magnetic dipole in y-direction, toroidal dipole in x-direction, electric quadrupole and magnetic quadrupole, respectively. Here, higher order dipoles are ignored because of the extremely smaller intensity on the scattered powers. We can see that the scattering power of magnetic dipole is the largest at 1115 nm, which proves it is a magnetic dipole resonance (MR). For the resonance at 1330 nm, the toroidal dipole is the dominating multipole contributions of scattering powers. It is approximately 2 times stronger than the electric dipole Px, about 7 times larger than the magnetic quadrupole QM and around 2 orders stronger than the electric quadrupole QE and the magnetic dipole My. It is noteworthy that the resonance at 1330 nm is an asymmetric dip, which may originate from the interference of the resonance of toroidal dipolar nature at 1330 nm with a broad sub-radiant, high-transmission background [33]. For the sake of simplicity, we call the resonance at 1330 nm as toroidal dipole resonance (TR). Here, the Q value of TR is calculated by fitting the transmission spectrum of Fig. 2(a) with the following Fano formula [34,35],
Fig. 2. (a) Transmission spectrum of the metasurface in the wavelength range of 1000 nm–1500 nm. (b) Five scattering powers of multipoles decomposition for the investigated metasurface, where Px, My, Tx, QE and QM are the electric dipole in x-direction, magnetic dipole in y-direction, toroidal dipole in x-direction, electric quadrupole and magnetic quadrupole, respectively. The log scale of y-axis is chosen in order to display more clearly the contribution results of the multipole decomposition. The vertical gray line labels the resonance wavelengths. The lattice periods are Px =750 nm and Py =1000 nm. 3
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 3. (a) E-field and (b) M-field enhancement profiles at the toroidal dipole resonance wavelength of 1330 nm when Px =750 nm and Py =1000 nm. The black arrows represent the directions of electric or magnetic fields.
I∝
(Fγ + ω − ω0 )2 (ω − ω0 )2 + γ 2
(11)
where F is the Fano parameter, γ and ω0 represent the width and position of the resonance. The Q value is calculated by ω0 /γ. For the case, the fitting parameter are F = 1, ω0 = 1418 rad/s, γ = 4.05 rad/s, respectively. The Q factor of TR is about 350. Fig. 3 shows the relative enhancements of the electric and magnetic nearfields at the x–y and y–z planes of the silicon bar, respectively. The black arrows represent the field vector distribution. The field enhancements are calculated at the wavelengths corresponding to the toroidal dipole resonance for Px =750 nm and Py =1000 nm. It is found that there are opposite circular displacement currents on two sides of the silicon bar simultaneously, each of them could generate a magnetic field, which upside is along +z axis, and -z axis for downside. These induced magnetic field patterns with low radiation into far filed can produce a circular magnetic moment confined in the volume of the silicon bar. Therefore, the head-to-tail magnetic moment connected leads to the formation of toroidal dipole moment along the x-axis [8,36]. In our metasurface, both electric and magnetic fields are confined inside the silicon bars (see Fig. 3), so each silicon bar can be seen as an effective waveguide [37–40], the intercoupling between the waveguides may play an essential role in the Q value of the TR. We firstly adjust the value of the lattice period in y-direction from 1000 nm to 720 nm. Fig. 4 shows the transmission spectra of the metasurface at four different lattice periods. We can see that both wavelength and bandwidth of MR remains mostly unaltered, but the wavelength of the TR slightly shifts from 1330 nm to 1308 nm. Meanwhile, the bandwidth of the TR has a dramatic decrease. To further clearly show the characters of Q value at TR, the Q value as a function of the Py for the all-dielectric metasurface is shown in Fig. 5. Here, Q value is also calculated by ω0 /γ. It is revealed that the Q value increases slowly, as the lattice period Py alters from 1000 nm to 850 nm, but rises quickly when Py changes from 850 nm to 720 nm. We can notice that the Q value reaches 63,000 when Py =720 nm, which is nearly 180 times stronger than that of Py =1000 nm. To study the collective oscillation in the metasurface, the electric field (x–y plane), magnetic field (y–z plane) and illustration of the toroidal dipole moments in multiple unit cells is given in Fig. 6 (Px = Py =750 nm). We referred to the toroidal dipole moment
Fig. 4. Transmission spectra of the metasurface at different lattice periods Py, where Px is fixed of 750 nm. The inset shows the magnified spectra of the TR. 4
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 5. The quality-factor dependence of TR on the lattice period Py, ranging from 1000 nm to 720 nm.
Fig. 6. The electric field (x–y plane), magnetic field (y–z plane) and illustration of the toroidal dipole moments in multiple unit cells. j (green arrows) represent the displacement currents and m (yellow arrows) represent the magnetic moments. T1 and T2 (claret arrows) represent intratoroidal dipole moment and inter-toroidal dipole moment, respectively. Small black arrows represent the directions of electric or magnetic fields (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.).
within the unit cell as the intra-toroidal dipole moment and the toroidal dipole moment between the unit cells as the inter-toroidal dipole moment. As shown in Fig. 6, there are two kinds of toroidal dipole moments: one is intra-toroidal dipole moments (named as T1), and the other is the inter-toroidal dipole moment (named as T2). For the intra-toroidal dipole moment, there are opposite circular displacement currents on two sides of one silicon bar simultaneously (such as j1 and j2), each of them could generate a magnetic field. These induced magnetic field patterns can produce a circular magnetic moment (intra-magnetic moment, such as m1) confined in the volume of the silicon bar. Therefore, the head-to-tail magnetic moment connected leads to the formation of intra-toroidal dipole moment along the x-axis [8,36], which has been verified in Fig. 3. The inter-toroidal dipole moment is produced by circular displacement currents of neighboring silicon bars along y-axis (such as j2 and j3). Interestingly, the circular displacement currents and the corresponding magnetic moments in the lower part of the silicon bar (such as j3) and upper part of the neighboring one (such as j2) are opposite to each other. The head-to-tail configuration of magnetic moment (inter-magnetic moment, such as m2) results in strong intercoupling between the neighboring silicon bars and indicates the excitation of inter-toroidal dipole moment oriented along the x-axis. In order to analyze the contribution of inter- and intra-toroidal dipole moments when decreasing the distance between the adjacent silicon bars, the distributions of the magnetic field in the y–z plane at the TR are illustrated in Fig. 7(a) ∼ 7(d) when Py =1000 nm, 900 nm, 800 nm and 720 nm, respectively. The intensity of inter-magnetic moment and intra-magnetic moment can reflect the intensity of inter-toroidal dipole moment and intra-toroidal dipole moment, respectively. We notice that the intensity of both inter- and intra-magnetic moment can be dramatically increased by decreasing the distance between the adjacent silicon bars. This means the inter- and intra-toroidal dipole moments be significantly excited by changing the lattice period in y-direction. The electric and magnetic field of the incident wave is significantly confined both inside and between the silicon bars as the toroidal dipole moments increase, resulting in achievements of the high-Q resonance of the metasurface [8,9,22]. Interestingly, the Q value at TR as a function of the Py keeps increased monotonically as given in Fig. 5, which means the inter-toroidal dipole moment and intra5
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
Fig. 7. (a), (b), (c), (d) Distributions of the magnetic field in the y–z plane at TR when Py =1000 nm, 900 nm, 800 nm and 720 nm, respectively. m1 is the intra-magnetic moment and m2 is the inter-magnetic moment.
one may not interfere with each other. We believe that the reason for this phenomenon may be that inter-toroidal dipole moment and intra-one produce in different locations, which is different from the excitation of anapole mode [24–27]. In the above works, the silicon bar arrays are supposed to embedded in air. However, the TR of our proposed metasurface can be also excited effectively with the presence of a substrate. In Fig. 8, the red solid line represents the transmission spectrum of the silicon bar arrays that are deposited on a semi-infinite quartz substrate (n = 1.46), and the black solid line represents the transmission spectrum of the silicon bars that is freestanding in air, in both cases, Px = Py =750 nm. Compared with the metasurface without substrate, we can see that the wavelength of the TR for the metasurface with a substrate exhibits a red-shift from 1310 nm to 1330 nm, and the Q value of the resonance decreases slightly from 6650 to 4140, because of the dielectric losses of the substrate. This means that the influence of a substrate on the Q factor of the toroidal resonance is much small. Besides, the Q-factor may be further enlarged by carefully designing the geometry parameters of silicon bars.
3. Conclusion In conclusion, we have presented and numerically investigated the simultaneous excitation of the intra-toroidal dipole moment
Fig. 8. Transmission spectra of silicon bar arrays deposited on a semi-infinite quartz substrate (red solid line) and disposed in air (black solid line) when Px = Py =750 nm, respectively. The inset shows the magnified spectrum of the TRs (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.). 6
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
and inter-toroidal dipole moment in all-dielectric metasurface consisting of silicon bar arrays in the near-infrared spectral region. Cartesian multipolar decomposition indicates that a narrowband toroidal resonance is observed in the metasurface. We found that the contribution of the weak free-space coupling of the inter-toroidal dipole and intra-toroidal dipole can be dramatically increased by decreasing the distance between the adjacent silicon bars. Thus, the electric and magnetic field of the incident wave is significantly confined both inside and between the silicon bars, resulting in achievements of high-Q toroidal resonance of the metasurface. Besides, we also demonstrated that the TR can still be effectively excited with a small decrease of Q value when the silicon metasurface is deposited on a quartz substrate. The proposed all-dielectric metasurface has the advantages of low loss and simple structure so that it may have wide applications in light-matter interactions, such as biochemical sensing, non-linear devices, slow light-based devices, and other optical devices. Acknowledgments The work was supported by the National Natural Science Foundation of China (Grant Nos. 61875179, 61875251 and 51704267) and the Natural Science Foundation of Zhejiang of China (Grant No. LY17E04002). References [1] I.B. Zel’Dovich, The relation between decay asymmetry and dipole moment of elementary particles, Sov. Phys. JETP 6 (1958) 1148. [2] V.M. Dubovik, L.A. Tosunian, V.V. Tugushev, Axial toroidal moments in electrodynamics and solid-state physics, Zh. Eksp. Teor. Fiz. 90 (1986) 590–605. [3] G. Thorner, J. Kiat, C. Bogicevic, I. Kornev, Axial hypertoroidal moment in a ferroelectric nanotorus: a way to switch local polarization, Phys. Rev. B Condens. Matter Mater. Phys. 89 (2014) 220103, , https://doi.org/10.1103/PhysRevB.89.220103. [4] I.I. Naumov, L. Bellaiche, H.X. Fu, Unusual phase transitions in ferroelectric nanodisks and nanorods, Nature 432 (2004) 737–740, https://doi.org/10.1038/ nature03107. [5] N.A. Spaldin, M. Fiebig, M. Mostovoy, The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect, J. Phys. Condens. Matter 20 (2008) 434203, , https://doi.org/10.1088/0953-8984/20/43/434203. [6] A. Ceulemans, L.F. Chibotaru, P.W. Fowler, Molecular anapole moments, Phys. Rev. Lett. 80 (1998) 1861–1864, https://doi.org/10.1103/PhysRevLett.80.1861. [7] Y. Yamaguchi, T. Kimura, Magnetoelectric control of frozen state in a toroidal glass, Nat. Commun. 4 (2013) 2063, https://doi.org/10.1038/ncomms3063. [8] T. Kaelberer, V.A. Fedotov, N. Papasimakis, D.P. Tsai, N.I. Zheludev, Toroidal dipolar response in a metamaterial, Science 330 (2010) 1510–1512, https://doi. org/10.1126/science.1197172. [9] Z. Dong, P. Ni, J. Zhu, X. Yin, X. Zhang, Toroidal dipole response in a multifold double-ring metamaterial, Opt. Express 20 (2012) 13065–13070, https://doi.org/ 10.1364/OE.20.013065. [10] L.Y. Guo, M.H. Li, Q.W. Ye, B.X. Xiao, H.L. Yang, Electric toroidal dipole response in split-ring resonator metamaterials, Eur. Phys. J. B. 85 (2012) 1–5, https:// doi.org/10.1140/epjb/e2012-20935-3. [11] Y. Fan, Z. Wei, H. Li, H. Chen, C.M. Soukoulis, Low-loss and high-Q planar metamaterial with toroidal moment, Phys. Rev. B 87 (2013) 115417, , https://doi.org/ 10.1103/PhysRevB.87.115417. [12] T.A. Raybould, V.A. Fedotov, N. Papasimakis, I. Kuprov, I.J. Youngs, W.T. Chen, D.P. Tsai, N.I. Zheludev, Toroidal circular dichroism, Phys. Rev. B 94 (2016), https://doi.org/10.1103/PhysRevB.94.035119. [13] S. Han, L. Cong, F. Gao, R. Singh, H. Yang, Observation of Fano resonance and classical analog of electromagnetically induced transparency in toroidal metamaterials, Ann. Phys.-Berlin 528 (2016) 352–357, https://doi.org/10.1002/andp.201600016. [14] L. Cong, Y.K. Srivastava, R. Singh, Tailoring the multipoles in THz toroidal metamaterials, Appl. Phys. Lett. 111 (2017) 081108, , https://doi.org/10.1063/1. 4993670. [15] Y. Fan, F. Zhang, N. Shen, Q. Fu, Z. Wei, H. Li, C.M. Soukoulis, Achieving a high-Q response in metamaterials by manipulating the toroidal excitations, Phys. Rev. A 97 (2018) 033816, , https://doi.org/10.1103/PhysRevA.97.033816. [16] M. Gupta, V. Savinov, N. Xu, L. Cong, G. Dayal, S. Wang, W. Zhang, N.I. Zheludev, R. Singh, Sharp toroidal resonances in planar terahertz metasurfaces, Adv. Mater. 28 (2016) 8206–8211, https://doi.org/10.1002/adma.201601611. [17] M. Gupta, Y.K. Srivastava, R. Singh, A toroidal metamaterial switch, Adv. Mater. 30 (2018), https://doi.org/10.1002/adma.201704845. [18] M. Gupta, Y.K. Srivastava, M. Manjappa, R. Singh, Sensing with toroidal metamaterial, Appl. Phys. Lett. 110 (2017) 121108, , https://doi.org/10.1063/1. 4978672. [19] B. Han, X. Li, C. Sui, J. Diao, X. Jing, Z. Hong, Analog of electromagnetically induced transparency in an E-shaped all-dielectric metasurface based on toroidal dipolar response, Opt. Mater. Express 8 (2018) 2197–2207, https://doi.org/10.1364/OME.8.002197. [20] Z. Liu, S. Du, A. Cui, Z. Li, Y. Fan, S. Chen, W. Li, J. Li, C. Gu, High-Quality-Factor Mid-Infrared toroidal excitation in folded 3D metamaterials, Adv. Mater. 29 (2017) 1606298, , https://doi.org/10.1002/adma.201606298. [21] B. Oeguet, N. Talebi, R. Vogelgesang, W. Sigle, P.A. van Aken, Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible, Nano Lett. 12 (2012) 5239–5244, https://doi.org/10.1021/nl302418n. [22] Y.W. Huang, W.T. Chen, P.C. Wu, V. Fedotov, V. Savinov, Y.Z. Ho, Y.F. Chau, N.I. Zheludev, D.P. Tsai, Design of plasmonic toroidal metamaterials at optical frequencies, Opt. Express 20 (2012) 1760–1768, https://doi.org/10.1364/OE.20.001760. [23] F. Huang, Z. Wang, X. Lu, J. Zhang, K. Min, W. Lin, R. Ti, T. Xu, J. He, C. Yue, J. Zhu, Peculiar magnetism of BiFeO3 nanoparticles with size approaching the period of the spiral spin structure, Sci. Rep. 3 (2013) 2907, https://doi.org/10.1038/srep02907. [24] P.C. Wu, C.Y. Liao, V. Savinov, T.L. Chung, W.T. Chen, Y.W. Huang, P.R. Wu, Y.H. Chen, A.Q. Liu, N.I. Zheludev, D.P. Tsai, Optical anapole metamaterial, ACS Nano 12 (2018) 1920–1927, https://doi.org/10.1021/acsnano.7b08828. [25] L. Wei, Z. Xi, N. Bhattacharya, H.P. Urbach, Excitation of the radiationless anapole mode, Optica 3 (2016) 799, https://doi.org/10.1364/OPTICA.3.000799. [26] S. Liu, Z. Wang, W. Wang, J. Chen, Z. Chen, High Q-factor with the excitation of anapole modes in dielectric split nanodisk arrays, Opt. Express 25 (2017) 22375–22387, https://doi.org/10.1364/OE.25.022375. [27] J.F. Algorri, D.C. Zografopoulos, A. Ferraro, B. Garcia-Camara, R. Beccherelli, J.M. Sanchez-Pena, Ultrahigh-quality factor resonant dielectric metasurfaces based on hollow nanocuboids, Opt. Express 27 (2019) 6320–6330, https://doi.org/10.1364/OE.27.006320. [28] S. Yang, Z. Liu, L. Jin, W. Li, S. Zhang, J. Li, C. Gu, Surface plasmon polariton mediated multiple toroidal resonances in 3D folding metamaterials, ACS Photonics 4 (2017) 2650–2658, https://doi.org/10.1021/acsphotonics.7b00529. [29] S. Xu, A. Sayanskiy, A.S. Kupriianov, V.R. Tuz, P. Kapitanova, H. Sun, W. Han, Y.S. Kivshar, Experimental observation of toroidal dipole modes in All-Dielectric metasurfaces, Adv. Opt. Mater. 7 (2019) 1801166, , https://doi.org/10.1002/adom.201801166. [30] A.A. Basharin, M. Kafesaki, E.N. Economou, C.M. Soukoulis, V.A. Fedotov, V. Savinov, N.I. Zheludev, Dielectric metamaterials with toroidal dipolar response, Phys. Rev. X 5 (2015) 011036, , https://doi.org/10.1103/PhysRevX.5.011036. [31] E.E. Radescu, G. Vaman, Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles, Phys. Rev. E 65 (2002) 046609, , https://doi.org/10.1103/PhysRevE.65.046609. [32] V. Savinov, V.A. Fedotov, N.I. Zheludev, Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials, Phys. Rev. B 89 (2014)
7
Optik - International Journal for Light and Electron Optics 201 (2020) 163502
X. Li, et al.
025112, , https://doi.org/10.1103/PhysRevB.89.205112. [33] N. Liu, T. Weiss, M. Mesch, L. Langguth, U. Eigenthaler, M. Hirscher, C. Soennichsen, H. Giessen, Planar metamaterial analogue of electromagnetically induced transparency for plasmonic sensing, Nano Lett. 10 (2010) 1103–1107, https://doi.org/10.1021/nl902621d. [34] S.H. Fan, Sharp asymmetric line shapes in side-coupled waveguide-cavity systems, Appl. Phys. Lett. 80 (2002) 908–910, https://doi.org/10.1063/1.1448174. [35] B. Luk’Yanchuk, N.I. Zheludev, S.A. Maier, N.J. Halas, P. Nordlander, H. Giessen, C.T. Chong, The Fano resonance in plasmonic nanostructures and metamaterials, Nat. Mater. 9 (2010) 707–715, https://doi.org/10.1038/NMAT2810. [36] K. Marinov, A.D. Boardman, V.A. Fedotov, N. Zheludev, Toroidal metamaterial, New J. Phys. 9 (2007), https://doi.org/10.1088/1367-2630/9/9/324. [37] M. Khorasaninejad, F. Capasso, Broadband multifunctional efficient Meta-Gratings based on dielectric waveguide phase shifters, Nano Lett. 15 (2015) 6709–6715, https://doi.org/10.1021/acs.nanolett.5b02524. [38] A. Arbabi, Y. Horie, A.J. Ball, M. Bagheri, A. Faraon, Subwavelength-thick lenses with high numerical apertures and large efficiency based on high-contrast transmitarrays, Nat. Commun. 6 (2015) 7069, https://doi.org/10.1038/ncomms8069. [39] M. Khorasaninejad, A.Y. Zhu, C. Roques-Carmes, W.T. Chen, J. Oh, I. Mishra, R.C. Devlin, F. Capasso, Polarization-Insensitive metalenses at visible wavelengths, Nano Lett. 16 (2016) 7229–7234, https://doi.org/10.1021/acs.nanolett.6b03626. [40] C. Sui, X. Li, T. Lang, X. Jing, J. Liu, Z. Hong, High Q-Factor resonance in a symmetric array of All-Dielectric bars, Appl. Sci. 8 (2018) 161, https://doi.org/10. 3390/app8020161.
8