Open Access
Add to BibSonomy
Abstract
Here we propose a metasurface consisting of symmetry-broken dielectric tetramer arrays, which can generate polarization-selective dual-band toroidal dipole resonances (TDR) with ultra-narrow linewidth in the near-infrared region. We found, by breaking the C4v symmetry of the tetramer arrays, two narrow-band TDRs can be created with the linewidth reaching ∼ 1.5 nm. Multipolar decomposition of scattering power and electromagnetic field distribution calculations confirm the nature of TDRs. A 100% modulation depth in light absorption and selective field confinement has been demonstrated theoretically by simply changing the polarization orientation of the exciting light. Intriguingly, it is also found that absorption responses of TDRs on polarization angle follow the equation of Malus’ law in this metasurface. Furthermore, the dual-band toroidal resonances are proposed to sense the birefringence of an anisotropic medium. Such polarization-tunable dual toroidal dipole resonances with ultra-narrow bandwidth offered by this structure may find potential applications in optical switching, storage, polarization detection, and light emitting devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The interaction of electromagnetic radiation with matter underpins some of the most important technologies today. Toroidal dipole (TD), as the primary component in a new independent family of elementary electromagnetic sources, was first introduced by Y. B. Zeldovich for parity violation explanation in the atomic nucleus [1]. To date, two sets of toroidal dipoles have been considered, termed magnetic (polar) toroidal dipole (MTD) and an electric (axial) toroidal dipole (ETD). The former is generated by poloidal electric currents flowing on the surface of a torus along its meridians, which can be represented as a set of magnetic dipoles (MDs) arranged head-to-tail forming a closed loop [2,3]. While the ETD is produced by the ring of the polarization current flowing along the toroidal direction. Like the toroidal vortices of light created recently [4,5], toroidal dipole resonances can provide a unusual manner for the enhanced light-matter interactions in perfect absorption [6,7], resonant forward scattering [8], harmonic generation [9], nanoscale source of lasing [10], spectroscopy and sensing [11–14], which has garnered the increasing interests. Although TDs exist in natural materials, their electromagnetic responses are very weak [15–17], which are usually masked by the much stronger effects of electric or magnetic multipoles. Enhancing and observing a clear signature of toroidal dipole resonance for strong light-matter interaction of nanostructures has still proven elusive in the past.
Metamaterials, artificial media periodically structured at the subwavelength scale to achieve the desirable electromagnetic functionality, served as a new platform for the observation of toroidal dipole resonances. Experimental signatures of a toroidal dipole response were first observed in the microwave dichroism spectra of chiral toroidal wire arrays on a Torus in 2009 [18], and it follows the observation of an isolated toroidal dipole absorption resonance in a metamaterial whose meta-atoms were formed by a ring-shaped arrangement of microwave resonators in 2010 [2]. Henceforth, these initial observations were followed by works aiming to further enhance the toroidal response and suppress the effects of competing electric and magnetic multipoles. To simplify the fabrication of three-dimensional toroidal metamaterials, quasi-planar designs were considered. For instance, split-ring resonators are replaced by pairs of bars [19] and disks [20], which still support toroidal resonances in THz frequencies. Also, toroidal dipole response was found in several simpler plasmonic systems, including plasmonic core-shell nanoparticles [21], spoof plasmonic two-dimensional groove metal arrays [22], plasmonic void oligomers [23], a silver circular V-groove array [24]. Nevertheless, due to the larger absorption and radiative loss in the metallic nanostructures, these plasmonic metasurfaces usually have a broad linewidth in spectral response of toroidal dipole resonance, which imposes a constraint on its applications for efficient sensing and light-matter interaction enhancement.
Low-loss nanostructured dielectrics replace the conduction current with low-loss displacement current [25–28], which make them a promising alternative to achieve the high-Q toroidal resonance. Recently, dielectric metasurfaces with the unit cell arrangements of several dielectric meta-atoms have been proposed to excite the strong toroidal responses. For example, periodic arrays of two pairs of high-index dielectric disks were used for the toroidal dipole based terahertz sensing [29]. By arranging two types of dielectric nanodisks into asymmetric quadrumer clusters, a strong axial toroidal response were obtained in NIR band [27]. More recently, a symmetric single-sized Si cuboids tetramer clusters were designed to realize a high-Q toroidal dipole resonance, which is based on the symmetry-protected bound state in the continuum (BIC) [26]. However, it shall be noted that most of the reported metasurfaces just support the TD resonance with a single band, while the dynamic control of the TD response with a large modulation depth and narrow-linewidth has proved elusive so far. Realization of the multiple-band narrow-linewidth resonators with a dynamically tunable TD response will benefit the development of various practical photonic and optoelectronic devices with high performance [17,30].
Here we propose a dielectric metasurface formed by arrays of asymmetric Si cuboids tetramer to produce dual-band toroidal dipole resonances in the NIR wavelengths theoretically. We found that the split dual-TD resonances are enabled by breaking the C4v symmetry of tetramer unit, which are characteristic of super-narrow linewidth and polarization-selective spatial confinement. Multipolar decomposition of scattering power and electro-magnetic field distribution calculations are performed to reveal the nature of toroidal dipole resonances. Our simulation results further show that the dual-band TD resonances can be selectively excited and dynamically controlled by simply tuning the polarization direction of the incoming light at resonant wavelength. These dual-narrow-band resonances are further proposed to sense the birefringence of an anisotropic medium. The dual-narrow-band toroidal dipole resonances with a polarization-tunable spectral response in this design pave the way for developing a variety of on-chip-integrated photonic and optoelectronic devices.
2. Structure design and principal
Figure 1(a) depicts the proposed metasurfaces, consisting of a periodic dielectric tetramer array on a 35 nm-thick Si3N4 film/Au mirror. The unit cell of tetramer arrays contains four Si cuboids, where the dimensions of each cuboid are initially set as the same, the cuboid width and height w1 = w2 = 260 nm, h = 200 nm, the separation of cuboids g = 120 nm, the period p = px = py = 800 nm. Here the dielectric constant of Au is adopted from experimental data of Palik [31], and the refractive index of Si and Si3N4 in the NIR wavelengths of interest is set as 3.42 and 1.93, respectively. The structure is illuminated by the linear-polarized plane waves impinging along the negative z-axis. Full-wave simulations were implemented by using the finite-difference time-domain (FDTD) method. In the electromagnetic modeling, the periodic boundary condition is applied to the unit cell along the x- and y-directions (left/right side), a perfectly matched layer (PML) condition is used in the z-direction (up/downside), and the spatial mesh grids are set as $\Delta x = \Delta y = \Delta z = 7\textrm{ nm}$, respectively.
Fig. 1. (a) Schematic of the proposed metasurface for generation of toroidal dipole resonance with narrow linewidth. (b) Top view of x-y sectional plane of unit cell. (c) The model of the single-port resonator. Structures are vertically illuminated by the x-polarized plane waves.
It has been demonstrated that toroidal dipole resonances can be excited in symmetric dielectric dimmer [32] or tetramer arrays [27,29], the elements of which have the same size. By breaking symmetry of the elements or the space group in periodic structures, a dark symmetry-protected bound states in the continuum (BICs) will turn into a leaky quasi-BIC mode with extremely high Q factor [33–36]. Other than changing the dimension or dielectric constant of the neighboring elements along the x- or y-direction [11,27,37], here we introduce the width difference Δ = (w2B- w2A)/2 of elements in two diagonal directions of the tetramer. In this condition, the unit of tetramer arrays can be viewed as an assembly of the two crosswise placed dimmers, which are denoted by group A and group B (Fig. 1(b)). As known, resonant frequency of TD relies on the geometry of dimmer, and the excitation efficiency of TD can be closely related to the polarization of incoming light [29,38,39]. Since the size of the dimmers in groups A and B are different, TD resonances should be selectively excited in dimmer A or B at different resonant frequencies, rendering dual TD excitations in this symmetry-broken tetramer array. For the tetramer arrays with C4v symmetry, TD resonances will merge in resonant frequency and manifest as a degenerate mode, because two dimmers have the same size.
Note that scattering behaviors of the resonators including the degenerate resonant modes can be well described by the coupled-mode theory (CMT) [40,41]. Thus, we can optimize and control absorption response for the multiple degenerate modes in single or two-port resonators. Here we consider the absorption behavior of a resonator that supports two resonances, which communicates with the outside via one identical port. The optical behavior of the resonator can be described by
Here $\alpha$ is the vector of complex amplitudes of resonant modes, t is the time. ${S_ + }$ gives the input amplitude at the port, and ${S_ - }$ present the output amplitude. C is a scattering matrix describing the scattering of the resonator in the absence of resonance. κ, d denotes the matrixes of coupling rate between the resonator and incoming/outgoing waves in two ports, and superscript T denotes the transpose of the matrixes. ${\omega _0}$ and ${\Gamma _i} = {\gamma _i} + {\delta _i}$ are the resonant frequency and the decay rate of the resonant mode, where ${\gamma _i}$ is the radiative decay rate of the i-th resonant mode and ${\delta _i}$ is its intrinsic decay rate due to absorption. For a single-port system, by eliminating $\alpha $ in Eq. (1), the reflectivity of the wave amplitude can be readily written asFor steady-state input from a single-port with unit amplitude at frequency ω, the absorptivity of the incoming light energy in the resonator reads
For the degenerate state with two resonant modes, the absorptivity of incident light can reach a maximum of 100%, when $\omega = {\omega _1} = {\omega _2}$, ${\gamma _1} + {\gamma _2} = {\delta _1} + {\delta _2}$. This is the condition for critical coupling of the degenerate modes in a single-port resonator. Particularly if γ1 = δ1 and γ2 = δ2 are satisfied simultaneously, two degenerate modes are both in the state of critical coupling at $\omega = {\omega _1} = {\omega _2}$, the perfect absorption is held. For a more common non-degenerate state, the radiation rate and dissipation rate (${\gamma _i},{\delta _i}$) of the resonances generally are associated with the modal field distributions, which can be tuned by modulating geometry and configuration of the resonator to reach a desired scattering (reflection/absorption) response [42,43]. It is also noted that radiation and dissipation properties of the resonant mode sometimes can be very sensitive to polarization direction of the excitation [44,45], which offers a practical way for dynamical control of the scattering response in resonant nanostructures.
3. Results and discussion
3.1 Dual toroidal dipole resonances in symmetry-broken tetramer arrays
Breaking the symmetry of nanostructures is one of effective ways to create the high-Q quasi-BIC modes, which enables a wealth of exotic optical phenomena [32,34,35,46]. Figure 2(a) illustrate the absorption spectra of the designed tetramer arrays as asymmetric parameter Δ under the normal incidence of x-polarized plane waves. For the structure with C4v symmetry (at Δ = 0), absorption spectra of the two split modes cross with each other, exhibiting a mode degenerate with notable absorption enhancement. By introducing Δ, the cuboid size of dimmer A reduces and that of dimmer B increases. C4v symmetry of tetramer arrays is broken. We see that the degenerate mode split into two branches of resonances, resonant frequencies of which demonstrate a clear linear-dependence on Δ. Generally, scattering responses of multimers are determined by the magnetic and electric resonances of individual cuboids and the electromagnetic coupling between the elements [17,27,32]. In Fig. 2(a), the resonant wavelength of these split modes as asymmetric parameter Δ can be described by
where k and w0 are the constant related to the geometric dispersion of the cuboid tetramer arrays (k = 1.3, w0 = 815.4 nm) and ‘${\pm} $’ denotes the high- and low-frequency mode, respectively. Note that here the geometric dispersions of these dual-TDRs are analogous to that of Mie resonances in the spherical dielectric particles [47].Fig. 2. (a) Absorption spectra of the proposed metasurface as the asymmetry parameter Δ under the normal illumination of x-polarized plane waves. (b) The absorption spectral lines at Δ = 0, 5, 13 nm, where insets display the Hz field distributions at the peaks in each panel, respectively.
To learn the mode properties in the proposed metasurface, the evolution of the absorption spectra and model field distributions at different Δ are given in Fig. 2(b), where parameters of the model are the same as those in section 2. Particularly, at Δ = 0, a single-peak resonance with a 100% perfect absorption can be observed at $\lambda $ = 1060 nm. The absorption peak exhibits a Lorentz spectral line-shape with a linewidth of 1.5 nm. At the peak, two sets of looped displacement currents can be observed in two cuboids of dimmers, corresponding to the degenerate toroidal dipole resonances. By further introducing the width difference (Δ = 5, 13 nm), cuboid size of dimmers A and B become different. As such, the looped displacement currents shall be selectively excited in one dimmer at the different resonant frequency. Regarding the structure with Δ = 5 nm, light absorptivity at two peaks are both 50% with the linewidth around 1.6 nm. Particularly, at the left peak, the coupled magnetic dipole resonances with a pair of looped currents and enhanced magnetic fields emerges in the cuboid of dimmer A; as for the right one, magnetic field distributions exhibit the similar patterns, but confined in the larger cuboids of dimmer B. In another word, the enhanced magnetic field can be selectively confined in dimmers A or B in these asymmetric tetramer arrays at different TD resonant frequencies. Moreover, Q factors of the TD resonances are calculated by $Q = \lambda /\Delta \lambda $, where $\lambda ,\,\,\Delta \lambda $ is the resonant wavelength and linewidth of the resonant mode. Here the Q factor of TD resonance for the structure at Δ = 0 nm is $0.706 \times {10^3}$, while that at Δ = 13 nm is $0.652 \times {10^3}$ and $0.673 \times {10^3}$, respectively.
For better understanding the nature of these resonant modes in the asymmetric metasurfaces, the scattering powers of different multipole for the symmetric and symmetry-broken tetramer arrays in the Cartesian coordinate were calculated, as shown in Figs. 3(a)–3(d). In calculations, the multiple pole moments of the electric dipole (ED), the magnetic dipole (MD), the toroidal dipole (TD), the electric quadrupole (EQ), and the magnetic quadrupole (MQ) are expressed as
Fig. 3. (a) The scattering powers of different multipoles for the symmetric structure at Δ = 0, and the scattered powers for x, y and z-components of the TD are indicated in (b). (c)-(d) The same as those in (a)-(b) but for the asymmetric structure at Δ = 13 nm. (e)-(f) Magnetic field distributions for the degenerate mode in the cross-sectional planes at z = 0 and x = 200 nm, respectively. (g)-(h) Magnetic field distribution for the dual non-degenerate modes. The white and black arrows herein represent the electric displacement and the magnetic field vectors.
For the symmetry-broken structure with Δ = 13 nm, we see that TD also dominates the multipole response in both of two split resonant modes (red solid in Figs. 3(c)-(d)). For the convenience of discussion, we refer to the toroidal dipole at λ = 1043 nm as TD1, while call that at λ = 1077 nm as TD2. At λ = 1043 nm, a pair of looped currents with the opposite-vortex can be observed in two cuboids of dimmer A (Fig. 3(g)), corresponding to the in-plane TD resonance in $\theta = {45^0}$. The displacement currents and enhanced magnetic field at λ = 1077 nm exhibit the similar patterns, while confined in the larger dimmer B, which is the TD excited in $\theta ={-} \textrm{ }{45^0}$. Here the in-plane TD moment vector for TD1 and TD2 ($\theta ={\pm} \textrm{ }{45^0}$) can be viewed as the superposition of the x- and y-components of TD resonance, which have the nearly equivalent magnitude in this symmetry-broken structure, as shown in Figs. 3(g)–3(h). It has been demonstrated that the near-field coupling between the individual Mie modes of the cuboids is capable of suppressing all standard multipoles [50,51], which makes electromagnetic scattering powers due to the resulting toroidal dipolar excitations the dominant in this regime. Meanwhile, it should be noted that TD resonances also can be excited in some the dielectric tetramer metasurfaces with the TD moment vector along the out-plane direction [50,52]. Structural parameters of the tetramers therein are displayed in Table 1. By comparison, we see that the direction of TD can be closely correlated to the scale size of height (H) and in-plane width (W) of components in the tetramer. Particularly, the TD resonances for the tetramers with larger aspect ratio (H/W) are typically along the out-plane direction, while those for the structures with small aspect ratio can be excited along the in-plane direction.
In short, the resonant vortex currents can be selectively excited in the symmetry-broken periodic Si tetramer arrays, producing a remarkable TD response in this resonant system. It is noteworthy that the excitation efficiency of TD is also associated with the polarization orientation of incoming light, which will be discussed in the following section 3.2.
Table 1. Structural Parameters of the Components in Several Tetramer Arrays in Published Works
3.2 Dependence of TD spectral response on polarization orientation of the light
It has been demonstrated that scattering responses of the resonant modes in optical nanostructures can be closely associated with the polarization state and electric vector direction of excitations [53–57]. To address the dependence of the TD response on the polarization of light, we investigated absorption responses in asymmetric tetramer arrays versus the polarization angle θ under the normal incidence of the linearly-polarized plane waves. Here polarization angle θ is defined as the angle between the direction of electric vector and the positive x-axis. Figure 4(a) displays the absorption spectra of the asymmetric metasurface (Δ = 13 nm) as a function of the polarization angle θ. The absorption spectral lines at θ = [450, 200, 00, -200, -450] and the corresponding |H| field distributions at the peaks are illustrated in Fig. 4(b). As can be seen, when the electric vector of light is aligned with one of diagonal directions ($\theta = {45^0}$), TD1 can be fully excited. It is manifested as a 100% perfect absorption with an enhanced magnetic field confinement in the cuboids of dimmer A (up panel in Fig. 4(b)). With the electric vector slowly rotated to the position $\theta ={-} {45^0}$, the absorption peak gradually decreases to zero and the resonant frequency of TD1 retains unchanged. As for the TD2 resonance, the evolution of absorption response and the modal field distributions on polarization angle θ exhibits the same way. But the polarization directions for perfect absorption and complete extinction are orthogonal to those of TD1 mode.
Fig. 4. (a) Simulated absorption spectra of the proposed metasurface at Δ = 13 nm as a function of polarization angle θ. (b) Absorption spectral lines at θ = 45°, 20°, 0°, -20°, -45° in (a), where the insets on left/right display the |H| filed distributions at the resonant peak of TD1 and TD2 modes. Absorption spectrum for the symmetric metasurface (Δ = 0 nm) is also indicated with dashed lines for comparison. (c)-(d) Peak absorption coefficient of the structure for TD1 and TD2 modes as a function of polarization angle θ, which are fitted and displayed in the black dashed lines. White arrows in insets indicate the direction of TD1 and TD2.
To quantitatively describe the excitation efficiency of the dual band TD resonances on the polarization direction, we plot the peak value of absorptivity with the polarization angle θ in Figs. 4(c)-(d). Here the excitation efficiency of TD as a function of polarization angle θ in this asymmetric metasurface can be described by
where P0 denotes the normalized absorption power of ingoing light. By using Eqs. (11)–(12), the fitted absorption coefficients are indicated in black dashed lines (Figs. 4(c)–4(d)). Intriguingly, it is demonstrated that the polarization dependence of TD absorption response in these asymmetric metasurfaces match perfectly with the Malus’ law in optics [58]. Essentially, the absorption responses for the dual-band TD resonances exhibit a distinct polarization matching effect, allowing for a 100% modulation depth of dual-channel light absorption in this regime. Additionally, TD resonances with the enhanced magnetic fields can be selectively excited in the dimmer A or dimmer B of these asymmetric tetramer arrays at the different resonant wavelengths, respectively. The unique polarization-tunable dual-band absorption response offered by the proposed metasurface will contribute to potential applications in polarization detection, sensing and light emission.3.3 Birefringence sensing based on the dual-band TD resonances
By virtue of the high Q factor and the larger mode volume, nanostructures with TD resonances have been utilized for refraction index sensing of the homogeneous isotropic medium [11,13,14,29]. Thanks to the dual-band TD resonances with ultra-narrow linewidth in the proposed structures, we propose a strategy to sense and distinguish the birefringence refractive indices no and ne and thickness for an anisotropic film. Figure 5(a) illustrate the proof-of-concept diagram of the birefringence sensor. It consists of an asymmetric metasurfaces (for example, Δ = 13 nm), which is covered by anisotropic thin film with thickness t. For simplicity, we assume that thin film material is uniaxial crystals with its optic axis along the x-direction. Figure 5(b) shows the refractive index ellipsoid of the uniaxial crystals, where no and ne denote the refractive indices for ordinary light and extraordinary light, respectively.
Fig. 5. (a) The diagram of structures for birefringence sensing, in which the metasurface is covered by an anisotropic media with uniaxial crystals. (b) The dispersion relation of the anisotropic film, and the optic axis is along the positive x-direction. (c)-(d) Birefringence indices sensing by resonance peak shifts of dual-band resonances of TD1 and TD2. Red dots are resonant peaks from the calculated absorption spectra and blue planes are 2D linear fitting results. (e)-(f) The resonant wavelengths as a function of ne at absorption peak of TD1 and TD2 modes with film thickness t = 40 nm, 60 nm, 80 nm and no = 1.32, which are the linearly fitted and displayed with solid lines. Other parameters in simulation are the same as those in Fig. 3(c).
Generally, spectral locations of resonant peaks are functions of the refractive indices and the thickness of the cladding layer, namely, λ1 = g1(no, ne, t), λ2 = g2(no, ne, t). Each resonant wavelength in each of the equations acts as an independent sensing channel, which supports the sensing of one physical quantity. Assuming that the equations are complete, the more channels we have, the more freedom and precise of the sensing we are able to achieve. Here we use the resonant wavelengths of dual-band TD resonances (TD1 and TD2) as independent channels to sense no and ne. Figures 5(c)–5(d) show the calculated resonant wavelength as functions of the sensing variables no and ne, which are well fitted through the following equations of ${\lambda _1} = 775.81 + 122.71{n_0} + 131.98{n_e}$, ${\lambda _2} = 834.70 + 114.43{n_0} + 113.14{n_e}$ and indicated with the blue planes. On basis of it, we also show the simulated resonant peak linear fitting of the refractive indices ne at different cladding thicknesses (t = 40 nm, 60 nm, 80 nm) in Figs. 5(e)–5(f). It can be seen that the resonant wavelength for both TD1 and TD2 changes remarkably for the different thickness of cladding layer. Consequently, the cladding thickness of anisotropy film should be in turn identified via measuring the spectral position of dual-channel TD resonances. Through resonant wavelength of dual-band TD resonances in the proposed metasurfaces, we can sense the birefringence refractive indices no, ne and thickness for a cladding layer of uniaxial crystals. It provides a practical way for the multiple-parameter optical sensing based on this resonant regime.
4. Conclusions
In summary, we designed a dielectric metasurface formed by arrays of asymmetric Si cuboid tetramer, which exhibits dual-band strong toroidal dipole resonances in the NIR region. Our simulation results show that the split dual-TD resonances can be excited by breaking the C4v symmetry of the tetramer unit, which are characteristic of super-narrow linewidth and selective spatial confinement. Multipolar decomposition of scattering power and electromagnetic distribution calculations are performed to confirm the physical mechanism of dual toroidal dipole resonances. It is also found that the dual-band TD resonances can be selectively excited and dynamically controlled by simply tuning the polarization direction of the incident light. Moreover, such dual-narrow-band resonances can be exploited to sense the birefringence properties of an anisotropic medium. Our findings could lead to many potential applications including ultra-compact optical switching, storage, encryption and light emitting devices.
Funding
National Natural Science Foundation of China (12204140, 62075053, 92050202, U20A20216); Fundamental Research Funds for the Central Universities (PA2020GDKC0024); Natural Science Foundation of Anhui Province (JZ2022AKZR0437).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. I. B. Zel’Dovich, “Electromagnetic Interaction with Parity Violation,” J. Exp. Theor. Phys. 6, 1184 (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]
3. 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]
4. C. Wan, Q. Cao, J. Chen, A. Chong, and Q. Zhan, “Toroidal vortices of light,” Nat. Photonics 16(7), 519–522 (2022). [CrossRef]
5. A. Zdagkas, C. McDonnell, J. Deng, Y. Shen, G. Li, T. Ellenbogen, N. Papasimakis, and N. I. Zheludev, “Observation of toroidal pulses of light,” Nat. Photonics 16(7), 523–528 (2022). [CrossRef]
6. H. Pan and H. Zhang, “Thermally tunable polarization-insensitive ultra-broadband terahertz metamaterial absorber based on the coupled toroidal dipole modes,” Opt. Express 29(12), 18081–18094 (2021). [CrossRef]
7. X. Chen and W. Fan, “Study of the interaction between graphene and planar terahertz metamaterial with toroidal dipolar resonance,” Opt. Lett. 42(10), 2034–2037 (2017). [CrossRef]
8. P. D. Terekhov, K. V. Baryshnikova, A. S. Shalin, A. Karabchevsky, and A. B. Evlyukhin, “Resonant forward scattering of light by high-refractive-index dielectric nanoparticles with toroidal dipole contribution,” Opt. Lett. 42(4), 835–838 (2017). [CrossRef]
9. G. Grinblat, Y. Li, M. P. Nielsen, R. F. Oulton, and S. A. Maier, “Efficient Third Harmonic Generation and Nonlinear Subwavelength Imaging at a Higher-Order Anapole Mode in a Single Germanium Nanodisk,” ACS Nano 11(1), 953–960 (2017). [CrossRef]
10. Y.-W. Huang, W. T. Chen, P. C. Wu, V. A. Fedotov, N. I. Zheludev, and D. P. Tsai, “Toroidal Lasing Spaser,” Sci. Rep. 3(1), 1237 (2013). [CrossRef]
11. Y. Huo, X. Zhang, M. Yan, K. Sun, S. Jiang, T. Ning, and L. Zhao, “Highly-sensitive sensor based on toroidal dipole governed by bound state in the continuum in dielectric non-coaxial core-shell cylinder,” Opt. Express 30(11), 19030–19041 (2022). [CrossRef]
12. Y. Wang, Z. Han, Y. Du, and J. Qin, “Ultrasensitive terahertz sensing with high-Q toroidal dipole resonance governed by bound states in the continuum in all-dielectric metasurface,” Nanophotonics 10(4), 1295–1307 (2021). [CrossRef]
13. X. Chen, W. Fan, and H. Yan, “Toroidal dipole bound states in the continuum metasurfaces for terahertz nanofilm sensing,” Opt. Express 28(11), 17102–17112 (2020). [CrossRef]
14. M. Gupta, Y. K. Srivastava, M. Manjappa, and R. Singh, “Sensing with toroidal metamaterial,” Appl. Phys. Lett. 110(12), 121108 (2017). [CrossRef]
15. V. A. Fedotov, K. Marinov, A. D. Boardman, and N. I. Zheludev, “On the aromagnetism and anapole moment of anthracene nanocrystals,” New J. Phys. 9(4), 95 (2007). [CrossRef]
16. K. Sawada and N. Nagaosa, “Gigantic enhancement of magnetochiral effect in photonic crystals,” Appl. Phys. Lett. 87(4), 042503 (2005). [CrossRef]
17. N. Papasimakis, V. A. Fedotov, V. Savinov, T. A. Raybould, and N. I. Zheludev, “Electromagnetic toroidal excitations in matter and free space,” Nat. Mater. 15(3), 263–271 (2016). [CrossRef]
18. N. Papasimakis, V. A. Fedotov, K. Marinov, and N. I. Zheludev, “Gyrotropy of a Metamolecule: Wire on a Torus,” Phys. Rev. Lett. 103(9), 093901 (2009). [CrossRef]
19. 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]
20. Z.-G. Dong, J. Zhu, X. Yin, J. Li, C. 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. W. Liu, J. Zhang, and A. E. Miroshnichenko, “Toroidal dipole-induced transparency in core–shell nanoparticles,” Laser Photonics Rev. 9(5), 564–570 (2015). [CrossRef]
22. S.-H. Kim, S. S. Oh, K.-J. Kim, J.-E. Kim, H. Y. Park, O. Hess, and C.-S. Kee, “Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons,” Phys. Rev. B 91(3), 035116 (2015). [CrossRef]
23. B. Ögüt, N. Talebi, R. Vogelgesang, W. Sigle, and P. A. van Aken, “Toroidal Plasmonic Eigenmodes in Oligomer Nanocavities for the Visible,” Nano Lett. 12(10), 5239–5244 (2012). [CrossRef]
24. J. Li, Y. Zhang, R. Jin, Q. Wang, Q. Chen, and Z. Dong, “Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection,” Opt. Lett. 39(23), 6683–6686 (2014). [CrossRef]
25. R. Chai, W. Liu, Z. Li, H. Cheng, J. Tian, and S. Chen, “Multiband quasibound states in the continuum engineered by space-group-invariant metasurfaces,” Phys. Rev. B 104(7), 075149 (2021). [CrossRef]
26. Y. Cai, Y. Huang, K. Zhu, and H. Wu, “Symmetric metasurface with dual band polarization-independent high-Q resonances governed by symmetry-protected BIC,” Opt. Lett. 46(16), 4049–4052 (2021). [CrossRef]
27. V. R. Tuz, V. V. Khardikov, and Y. S. Kivshar, “All-Dielectric Resonant Metasurfaces with a Strong Toroidal Response,” ACS Photonics 5(5), 1871–1876 (2018). [CrossRef]
28. J. Hu, Y. Xiao, L.-M. Zhou, X. Jiang, W. Qiu, W. Fei, Y. Chen, and Q. Zhan, “Ultra-narrow-band circular dichroism by surface lattice resonances in an asymmetric dimer-on-mirror metasurface,” Opt. Express 30(10), 16020–16030 (2022). [CrossRef]
29. X. Chen and W. Fan, “Ultrahigh-Q toroidal dipole resonance in all-dielectric metamaterials for terahertz sensing,” Opt. Lett. 44(23), 5876–5879 (2019). [CrossRef]
30. C. Zhou, S. Li, Y. Wang, and M. Zhan, “Multiple toroidal dipole Fano resonances of asymmetric dielectric nanohole arrays,” Phys. Rev. B 100(19), 195306 (2019). [CrossRef]
31. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1997).
32. Y. He, G. Guo, T. Feng, Y. Xu, and A. E. Miroshnichenko, “Toroidal dipole bound states in the continuum,” Phys. Rev. B 98(16), 161112 (2018). [CrossRef]
33. V. A. Fedotov, M. Rose, S. L. Prosvirnin, N. Papasimakis, and N. I. Zheludev, “Sharp Trapped-Mode Resonances in Planar Metamaterials with a Broken Structural Symmetry,” Phys. Rev. Lett. 99(14), 147401 (2007). [CrossRef]
34. K. Koshelev, S. Lepeshov, M. Liu, A. Bogdanov, and Y. Kivshar, “Asymmetric Metasurfaces with High-Q Resonances Governed by Bound States in the Continuum,” Phys. Rev. Lett. 121(19), 193903 (2018). [CrossRef]
35. T. Shi, Z.-L. Deng, G. Geng, X. Zeng, Y. Zeng, G. Hu, A. Overvig, J. Li, C.-W. Qiu, A. Alù, Y. S. Kivshar, and X. Li, “Planar chiral metasurfaces with maximal and tunable chiroptical response driven by bound states in the continuum,” Nat. Commun. 13(1), 4111 (2022). [CrossRef]
36. B. Meng, J. Wang, C. Zhou, and L. Huang, “Bound states in the continuum supported by silicon oligomer metasurfaces,” Opt. Lett. 47(6), 1549–1552 (2022). [CrossRef]
37. X. Liu, J. Li, Q. Zhang, and Y. Wang, “Dual-toroidal dipole excitation on permittivity-asymmetric dielectric metasurfaces,” Opt. Lett. 45(10), 2826–2829 (2020). [CrossRef]
38. L.-Y. Guo, M.-H. Li, X.-J. Huang, and H.-L. Yang, “Electric toroidal metamaterial for resonant transparency and circular cross-polarization conversion,” Appl. Phys. Lett. 105(3), 033507 (2014). [CrossRef]
39. A. S. Kupriianov, Y. Xu, A. Sayanskiy, V. Dmitriev, Y. S. Kivshar, and V. R. Tuz, “Metasurface Engineering through Bound States in the Continuum,” Phys. Rev. Applied 12(1), 014024 (2019). [CrossRef]
40. H. A. Haus, Waves and fields in optoelectronics (Prentice-Hall, 1984).
41. S. Wonjoo, W. Zheng, and F. Shanhui, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]
42. X. Wang, J. Duan, W. Chen, C. Zhou, T. Liu, and S. Xiao, “Controlling light absorption of graphene at critical coupling through magnetic dipole quasi-bound states in the continuum resonance,” Phys. Rev. B 102(15), 155432 (2020). [CrossRef]
43. J. Wang, A. Chen, Y. Zhang, J. Zeng, Y. Zhang, X. Liu, L. Shi, and J. Zi, “Manipulating bandwidth of light absorption at critical coupling: An example of graphene integrated with dielectric photonic structure,” Phys. Rev. B 100(7), 075407 (2019). [CrossRef]
44. L. Meng, D. Zhao, Q. Li, and M. Qiu, “Polarization-sensitive perfect absorbers at near-infrared wavelengths,” Opt. Express 21(S1), A111–A122 (2013). [CrossRef]
45. W. Wang, L. V. Besteiro, T. Liu, C. Wu, J. Sun, P. Yu, L. Chang, Z. Wang, and A. O. Govorov, “Generation of Hot Electrons with Chiral Metamaterial Perfect Absorbers: Giant Optical Chirality for Polarization-Sensitive Photochemistry,” ACS Photonics 6(12), 3241–3252 (2019). [CrossRef]
46. F. Wu, J. Wu, Z. Guo, H. Jiang, Y. Sun, Y. Li, J. Ren, and H. Chen, “Giant Enhancement of the Goos-Hanchen Shift Assisted by Quasibound States in the Continuum,” Phys. Rev. Applied 12(1), 014028 (2019). [CrossRef]
47. R. Yahiaoui, K. Hanai, K. Takano, T. Nishida, F. Miyamaru, M. Nakajima, and M. Hangyo, “Trapping waves with terahertz metamaterial absorber based on isotropic Mie resonators,” Opt. Lett. 40(13), 3197–3200 (2015). [CrossRef]
48. 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]
49. 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]
50. 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]
51. X. Chen, W. Fan, X. Jiang, and H. Yan, “High-Q Toroidal Dipole Metasurfaces Driven By Bound States in the Continuum for Ultrasensitive Terahertz Sensing,” J. Lightwave Technol. 40(7), 2181–2190 (2022). [CrossRef]
52. Z. Li, T. Wu, and X. Zhang, “Tailoring toroidal and magnetic dipole excitations with the same dielectric structure,” Opt. Lett. 44(1), 57–60 (2019). [CrossRef]
53. C. Huang, C. Zhang, S. Xiao, Y. Wang, Y. Fan, Y. Liu, N. Zhang, G. Qu, H. Ji, J. Han, L. Ge, Y. Kivshar, and Q. Song, “Ultrafast control of vortex microlasers,” Science 367(6481), 1018–1021 (2020). [CrossRef]
54. N. Yu and F. Capasso, “Flat optics with designer metasurfaces,” Nat. Mater. 13(2), 139–150 (2014). [CrossRef]
55. W. Liu, L. Mei, Y. Li, L. Yu, Z. Lai, T. Yu, and H. Chen, “Controlling the spin-selective absorption with two-dimensional chiral plasmonic gratings,” Opt. Lett. 44(23), 5868–5871 (2019). [CrossRef]
56. R. Gordon, A. G. Brolo, A. McKinnon, A. Rajora, B. Leathem, and K. L. Kavanagh, “Strong Polarization in the Optical Transmission through Elliptical Nanohole Arrays,” Phys. Rev. Lett. 92(3), 037401 (2004). [CrossRef]
57. L. Zhao, X. Yang, Q. Niu, Z. He, and S. Dong, “Linearly thermal-tunable near-infrared ultra-narrowband metamaterial perfect absorber with low power and a large modulation depth based on a four-nanorod-coupled a-silicon resonator,” Opt. Lett. 44(15), 3885–3888 (2019). [CrossRef]
58. E. Hecht, Optics, 5th ed. (Pearson Education, 2017).
