Open Access
Add to BibSonomy
Abstract
A novel terahertz nanofilm sensor consisting of toroidal dipole bound states in the continuum (TD-BIC) inspired Fano resonance metasurface is proposed and investigated, which exhibits both the TD character and BIC feature. When the mirror symmetry of the unit cell was broken, the TD resonance was excited and demonstrated by anti-aligned magnetic dipoles and calculated scattering powers and the BIC mode was verified with the quality factor satisfying the inverse square law. Combined with the amplitude difference referencing technique, the TD-BIC inspired Fano resonance was utilized for nanofilm sensing at THz frequencies for the first time. Simulation results show that the amplitude difference can be easily observed by comparing the resonance frequency shift under difference thicknesses of germanium overlayer. Moreover, by coating with a 40 nm-thick analyte overlayer, the sensitivity of amplitude difference can achieve 0.32/RIU, which is a significant value and more suitable for sensing nanofilm analytes than the traditional frequency shift method. These advantages make our proposed structure have potential applications in sensing nanofilm analytes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Toroidal dipoles (TDs), one of the fundamental electromagnetic excitations produced by currents flowing on the surface of a torus, were firstly predicted by Zel’dovich for parity violation explanation in atomic nucleus [1], and then explored in atomic, molecular and nuclear physics, classical electrodynamics and solid state physics [2,3]. With the characteristics of non-radiating configuration and electromagnetic fields confinement, many interesting physical phenomena such as nonreciprocal refraction, circular dichroism, and harmonic generation can be realized by TDs [4,5]. Since the naturally existing TDs are not considered in traditional multipoles expansion and classical electrodynamics [6] and masked by much stronger electric and magnetic multipoles, the characterization and detection of TDs is difficult, which has been neglected long time until experimentally demonstrated in three-dimensional metamaterials (MMs) at microwave regime [7]. MMs are periodically arranged subwavelength structure with many exotic phenomena such as super lensing, negative refractive index and perfect absorption, which attract enormous interest in the past few decades [8]. Moreover, its resonance frequency can be engineered from radio to terahertz (THz) and even near-infrared [7,9,10], strengthening the adaptability of MMs devices. However, the fabrication of three-dimensional toroidal MMs is usually difficult due to the restriction of micromachining technique. The planar toroidal metasurfaces have received growing attention with easy fabrication, high quality (Q) resonance and sufficiently field confinement, which are highly sensitive to the change of surrounding medium and suitable for sensing applications [11,12]. For example, a metallic metasurface consisting of square split ring resonators (SRRs) with the sharp TD resonance was reported [13] and a strong toroidal response was introduced and demonstrated by engineering asymmetric nanoparticle quadrumers in the all-dielectric metasurface [14]. For ultrasensitive sensors, the high Q resonance with extremely narrow bandwidth is very important and necessary, which indicates the lower rate of energy losses and supports strong interaction between electromagnetic waves and analytes [15]. With the radiation losses suppressed, the high Q-factor can be achieved by Fano resonance with an asymmetric lineshape and sharp spectral profile [16]. Specially, one effective approach to achieve extremely high Q Fano resonance is based on the bound states in the continuum (BIC) [17], a localized state with zero linewidth that is embedded in the continuum, where the Q-factor going infinity due to the resonance uncoupled to the free space radiation [18]. BIC was originally proposed by Friedrich and Wintgen in quantum mechanics, and then extended to acoustics, hydrodynamics, and optics [19]. In general, except for the nonsymmetry-protected BIC [20], the ideal BIC state is symmetry-protected and not observable in real systems, which can be realized as quasi-BIC (supercavity) mode with resonance linewidth and Q-factor become finite [21–23]. Specifically, the quasi-BIC mode was experimentally demonstrated in metasurface formed by dimer rod arrays with different rod sizes in the unit cell [24] and in all-dielectric metasuface with the supercavity resonance mode in the vicinity of the BIC mode [25]. Moreover, the quasi-BIC mode has been reported in symmetry broken SRRs with high Q and giant enhancement of fields for many useful functionalities like lasing and biosensing [26,27].
With the fingerprint spectral of materials such as DNA, proteins, and explosives exist at THz regime and the non-invasive, non-destructive characteristics of THz spectroscopy [28], THz waves and THz metasurfaces sensors are highly effective tool for chemical and biological sensing [29–31]. Many MMs and metasurfaces possessing sharp spectral features like Fano, quadrupole, and toroidal resonances have been demonstrated for refractive index sensing with the resonance frequency shift [32–34]. Generally, the fringing fields of MMs play a key role in sensing and extend to about few micrometers, thus needing a large thickness of analytes [32,33]. Once decreasing the analyte thickness to few nanometers, the frequency shift becomes minuscule and sensing nanometer dielectric films is challenging due to the reduced strength of light-matter interaction and the large discrepancy of THz wavelength and tiny quantity of analytes [35]. Besides, for conventional terahertz time-domain spectroscopy (THz-TDS) system, such a small frequency shift could only be measured with extremely sharp resonance features and high spectral resolution in frequency domain to discern close points, which means the frequency shift method become less effective for extremely low analyte volumes. Recently, an advanced sensing data analysis method was proposed for sensing nanofilm analytes at THz frequencies, which is based on the change of the transmission amplitude [36].
In this paper, the physical mechanism of toroidal dipole bound states in the continuum (TD-BIC) inspired Fano resonance is investigated comprehensively in THz metasurface composed of double SRRs, which exhibits TD character and BIC feature. With breaking mirror symmetry of the unit cell, the TD resonance is excited and demonstrated by anti-aligned magnetic dipoles and calculated scattering powers. Similarly, the resonance changes from Fano to BIC mode and the Q-factors obey the inverse square law with the asymmetry parameter decreasing, demonstrates the BIC feature. To our knowledge, it is the first time that the TD-BIC inspired Fano resonance is utilized for nanofilm sensing at THz frequencies. By using the amplitude difference referencing technique, simulation results show the amplitude difference can be easily observed comparing with the frequency shift under difference thicknesses of germanium (Ge) overlayer. Besides, with coating 40 nm-thick analytes overlayer, the sensitivity of amplitude difference can achieve 0.32/RIU, which is higher than reported THz BIC metasurfaces sensors [36,37]. These advantages make such metasurface feasible to use in biological and chemical nanofilm analytes sensing.
2. Structure design
Figure 1(a) shows the schematic view of the proposed THz sensor, which consists of unit cell arrays deposited on a 25 µm-thick flexible cyclic olefin copolymer substrate, where its refractive index n = 1.53 and loss tangent tan δ = 0.0006 [38]. Figure 1(b) shows the top view of the unit cell structure, which contains two SRRs and constructed by 200 nm-thick metal aluminum with conductivity σdc = 3.56 × 107 Sm−1 [33]. The geometric parameters of the unit cell are period P = 64 µm, length L = 50 µm, width w = 4 µm, spilt gap g1 = g2 = 3 µm, middle distance d = 4 µm and asymmetry parameter δ = 13 µm, respectively. Specially, the low refractive index and losses of the substrate allows strong local electric fields confinement in the split gaps of the resonator and thus further enhances the sensitivity of the metasurface sensor in comparison with the earlier sensing demonstrations with high index substrates [30]. To study the proposed structure and the interaction between TD-BIC inspired Fano resonance and analytes, the calculations were performed by a commercially available electromagnetic full-wave simulation software computer simulation technology microwave studio based on finite integration method. In the simulation, the unit cell boundary conditions were applied in x- and y-directions to characterize the periodic structure, and the open boundary condition was employed along the z-direction in the free space. THz waves were normal incidence with electric field along the x-direction. The calculations were carried out by the frequency domain solver and tetrahedral mesh type. At least 4 mesh steps per wavelength were used to ensure the accuracy of the calculated results.
Fig. 1. (a) Schematic view of the proposed THz metasurface sensor, where THz waves are normal incidence with E-field along x-direction and the analyte is ultrathin nanofilm. (b) Top view of the unit cell with structure parameters are P = 64 µm, L = 50 µm, w = 4 µm, g1 = 3 µm, g2 = 3 µm, d = 4 µm, and δ = 13 µm.
3. Results and discussions
At first, the transmission spectrum of the proposed structure is investigated with the asymmetry parameter δ = 13 µm. In Fig. 2(a), one can observe a sharp resonance dip at 1.2466 THz and the transmission profile with dip/peak pair shows a distinct asymmetric Fano lineshape [39]. Generally, in scattering problems, resonant states of a system manifest themselves as resonant enhancement at real frequencies. Since the energy leaks out from the scatter and gets dissipated, each specific resonant state is characterized by a complex frequency ω = Re(ω) + iIm(ω), with the position at the real axis Re(ω) and a finite lifetime τ = 1/γ = −1Im(ω), where γ is the decay rate and determines the resonance linewidth at its half-maximum [20]. Moreover, a resonance is usually described by the Q-factor and defined as Q = −Re(ω)/2Im(ω) = ω0/2γ, where the factor 2 occurs in the denominator due to the energy decays with the decay rate 2γ [20]. In addition, for the Q-factor of the asymmetric lineshape resonance, it can be extracted by a Fano model with the simulated transmission intensity spectrum fitted to a typical Fano formula and estimated as Q = ω0/2γ, where ω0 is the resonance frequency and γ is the overall damping rate [22,40]. The overall damping rate can be obtained by fitting the transmission spectrum with a typical Fano formula given by:
where a1, a2, and b are real constant numbers. Thus, the Q-factor of the transmission spectrum can be calculated as 143, which is a high Q value in THz metallic metasurfaces. To qualitatively study the physical mechanism of this sharp Fano resonance, the distributions of surface current, electric field, and magnetic field are simulated and analyzed in-depth. In Fig. 2(b), the surface currents are distributed with opposite flowing directions along metallic loops of unit cell, which form two closed rectangular loops (blue arrows) and generating two anti-aligned magnetic dipoles. It has been reported that the TD Fano resonance can be excited by a set of anti-parallel magnetic dipoles, where the magnetic dipole weakly couples to the free space and arises as a high Q resonance [33]. For the electric field distribution, as shown in Fig. 2(c), one can observe two pairs of opposite electric charges gathering around the split gap at left and right SRRs, which demonstrates the electric dipole can be excited and the direction is consistent with the direction of surface current (along − y-direction). In Fig. 2(d), the magnetic field m is induced by surface current and distributed around the left and right SRRs with opposite directions. Subsequently, the induced magnetic field wreathing around the middle gap of unit cell and forms the closed magnetic fields, thus generating the TD response along + y-direction. Specially, at this toroidal dipole resonance, with the destructive interference in the far-field, the electric dipole and toroidal dipole coherently oscillate and radiating suppressed [41], thus resulting in the sharp resonance. Furthermore, to quantitatively analyze the role of TD resonance in forming this high Q resonance, we perform the multipoles expansion of the induced current density extracted from unit cell and take into account five strongest terms, namely, electric dipole P, magnetic dipole M, toroidal dipole T, electric quadrupole Qe, and magnetic quadrupole Qm [42]. With the unit cell of metasurface sufficiently smaller than the wavelength of incident radiation, it allows us to replace the sum over the unit cell with an integral over the array area and a Taylor expansion in Cartesian coordinate can be applied for scattering analysis. In addition to electric and magnetic dipoles, quadrupoles, the expansion in Cartesian coordinate allows calculating the toroidal moments [43], which are higher-order Taylor expansions contributing to lower order spherical harmonics, indicating there is no toroidal multipole in the spherical multipoles [44]. Then, the scattering powers of different multipoles are calculated by summing the contributions of dipoles scattering from all unit cells in Cartesian coordinate as [43]:Fig. 2. (a) Transmission spectrum of the proposed structure. The distributions of (b) surface current (c) electric field and (d) magnetic field at the resonance frequency. (e) The calculated normalized powers scattered by different multipoles of the metasurface.
For symmetry-protected BIC, it has infinite Q-factor and can be transformed into quasi-BIC by breaking symmetry of the unit cell, leading to sharp resonance response. With the mechanism of BIC and excitation of quasi-BIC mode, one can control the radiation damping rate and engineer the Q-factor of resonance. So, it is important to investigate the physical mechanism of symmetry-protected BIC and quasi-BIC resonance in this metasurface. The transmission spectra with different asymmetry parameters δ are simulated in Fig. 3(a). Clearly, for symmetric structure with δ = 0, no transmission dip can be observed and the resonance linewidth vanishes, where the symmetry-protected BIC state can be supported. With the asymmetry parameter increasing and the mirror symmetry broken, the transmission spectra show distinct Fano resonance feature and the resonant frequency red shifts with the linewidth increasing. It means the BIC state is unstable against the perturbation with breaking mirror symmetry, which induces leakage of BIC and leads to quasi-BIC resonance mode. Moreover, the distributions of surface current for BIC and quasi-BIC with δ = 0 µm and δ = 4 µm are simulated to understand the transition from bound state to quasi-BIC in Figs. 3(b) and 3(c), as marked with circles in Fig. 3(a). For symmetry-protected BIC in Fig. 3(b), on the one hand, the surface current is really very weak, which indicates it hardly coupling to the free space radiation and is completely confined as a bound state without leakage channel. The other hand, the very weak in-phase collective currents on the top and bottom arms of resonators with the same direction of incident THz waves come from the dipole resonance of the symmetric structure. With breaking mirror symmetry (e.g. δ = 4 µm), as shown in Fig. 3(c), the surface current is strongly excited and the orientation (blue arrows) as well as magnitude are rearranged on the surface of resonators, which indicates the bound state couples to the incident radiation thus giving rise to quasi-BIC resonance. In Fig. 3(d), the Q-factors with different asymmetry parameters are also investigated, where the Q-factors are extracted from the simulated transmission spectra of the ideal metasurface (composed of a perfect electrical conductor (PEC) and lossless substrate) and the loss metasurface (composed of metallic Al and loss substrate) structures. Obviously, the Q-factor tends to infinity at δ = 0 µm due to no coupling of the mode to the free space and demonstrates the symmetry-protected BIC state for both structures. With breaking mirror symmetry by moving the right split gap of SRR away from x-axis, the BIC state transforms into the quasi-BIC mode and the Q-factors decrease gradually with asymmetry parameters increasing. The Q-factors of the ideal metasurface are significantly larger than those of the loss metasurface owing to their zero ohmic and substrate losses. In addition, for the ideal metasurface, the dependence of the Q-factors on the asymmetry parameter are investigated and can be fitted by the inverse quadratic law (Qrad ∝ α−2) very well, as the blue solid line shown, which is consistent with the results reported by Koshelev, et al. and Cong, et al. [21,22] and demonstrates this resonance is BIC inspired. Hence, the unit cell with breaking mirror symmetry is necessary to obtain a sharp quasi-BIC resonance where the Q-factor and linewidth can be controlled by the asymmetry parameter.
Fig. 3. (a) Transmission spectra of the proposed structure with different asymmetry parameters. (b)-(c) Surface current distributions with asymmetry parameter δ = 0 and 4 µm, respectively. (d) The Q-factor of quasi-BIC for ideal (red circle) and loss (black square) metasurface with different asymmetry parameters. Blue solid line shows theoretical fitting using the inverse square function with asymmetry parameters.
In addition to breaking mirror symmetry of the unit cell, tilting the incident wave vector is another way to couple the bound state to the normal incidence and explain the symmetry-protected BIC intuitively [22]. In order to facilitate simulation and analysis, the structure is considered as the ideal metasurface composed of PEC and lossless substrate with asymmetry parameter δ = 0. By sweeping the incident angle θ relative to the resonator plane in Fig. 4(a), the simulations are performed. Obviously, for normal incidence (θ = 0), no resonance features can be observed and the Q-factor tends to infinite in Fig. 4(b), which verify the BIC state. To quasi-BIC mode, a general method to illustrate it is to trace the diverging trajectory of the Q-factor. In Fig. 4(b), as the incident angle increasing, the Q-factors decrease gradually and reveal a trend to diverge to infinity at θ = 0, which manifest the quasi-BIC features. Therefore, with breaking mirror symmetry of the unit cell or tilting the incident angle, the symmetry-protected BIC and quasi-BIC mode can be demonstrated clearly.
Fig. 4. (a) Schematic diagram of the incident angle spectral analysis with different incident angle θ, where the asymmetry parameter is zero. (b) The calculated Q-factor of the BIC inspired resonance with different incident angles, where the Q-factor tends to infinity indicating the BIC state in the ideal metasurface.
For sensing application based on Fano resonance, the Q-factor and resonant intensity I (defined as the difference between the maximum and minimum transmission amplitude of resonance) are two particularly important parameters. Generally, in MMs or metasurfaces, the Fano lineshape can be optimized by decreasing the structural asymmetry to give a high Q-factor, whereas the optimization often leads to low resonance intensity and limits the comprehensive performance parameter Q × I [45,46], making the Fano spectrum difficult to be measured. Hence, it is important to determine the optimal asymmetric parameter which excites the high Q resonance with strong resonant intensity. As shown in Fig. 5(a), the value of Q-factor and resonant intensity are calculated with different asymmetry parameters. As the asymmetry parameter increasing, the transmission curves become broader, which results in the Q-factor decreasing and resonant intensity increasing. For the comprehensive performance parameter Q × I, as shown in Fig. 5(b), it increases firstly and then saturated with asymmetry parameter increasing to 13 µm. So, the asymmetry parameter δ = 13 µm is the optimized value and chosen as the TD-BIC inspired Fano resonance for sensing application.
Fig. 5. (a) The Q-factor and resonant intensity of transmission spectra with different asymmetry parameters. (b) The variation of Q × I with different asymmetry parameters.
Most importantly, the sensing performance of the proposed structure is the main focus of our study. By coating Ge layer on the metasurface, the simulations are performed and the sensing performances are analyzed by using the amplitude difference referencing technique. The Ge layer is chosen as the analyte because of its high dielectric constant (ɛ = 16) which results in a relatively large resonance frequency shift [37]. The proposed amplitude difference referencing technique is evaluated where the transmission amplitude of the metasurface with the analyte was subtracted from the transmission amplitude without the analyte [36,37]. Specifically, two sets of data are simulated including uncoated metasurface (without analytes) and coated metasurface with analytes depositing on the surface of metasurface, where the transmission spectrum of the uncoated metasurface is set as reference signal. In Fig. 6(a), the resonance frequency of the metasurface without Ge overlayer (black curve) is 1.2466 THz. After 7 nm Ge layer coated on the metasurface, owing to the alteration in the dielectric environment of the resonators, the resonance frequency (red line) is red shifted by 2.2 GHz to 1.2444 THz, which shows a quite low frequency shift. With increasing Ge thickness to 20 nm and 40 nm, the resonance shows red shift of 3.8 GHz and 5.9 GHz, respectively. The increased analyte thickness results in an enhanced light-matter interaction thus leading to the red shift of the TD-BIC inspired Fano resonance. Moreover, it is important to note that the red shift of the Fano resonance is very small and could be only discerned and measured by extremely high resolution THz-TDS. Therefore, the conventional frequency shift method becomes inefficient for sensing the nanofilm analytes at THz frequencies. To overcome the difficulty in sensing nanofilm analytes using the conventional THz-TDS system, the amplitude difference referencing technique to the same analyte is investigated and the performance as shown in Fig. 6(b). Clearly, for 7 nm thick Ge overlayer (red curve), the peak-to-peak difference of transmission amplitude (ΔT) is 49.38%, which is a significant value that can be easily measured and detected in a noisy environment using THz-TDS [47], illustrating the effectiveness of this method for ultrasensitive sensing of nanofilm analytes. With the thickness of analyte increasing to 20 nm and 40nm, the amplitude difference increases continuously and the values of 75.05% and 97.01% can be achieved, respectively, as shown in Fig. 6(b). The simulation results reflect the potential of using such a referencing technique especially in the case where the analyte thickness is in the nanoscale range.
Fig. 6. (a) Transmission spectra of the metasurface without and with 7, 20, and 40 nm Ge overlayer. (b) Transmission amplitude difference with 7, 20, and 40 nm Ge overlayer. (c) Transmission spectra of the metasurface coated with 40 nm-thick analyte with different refractive indices. The inset shows the resonant frequency shift with different refractive indices of analyte overlayer. (d) Transmission amplitude difference with different refractive indices of analyte, where its thickness fixed as 40 nm. The inset in (d) shows the peak-to-peak transmission amplitude difference |ΔT| with different refractive indices of analyte.
Furthermore, the sensitivity of the proposed metasurface by varying the refractive index of 40 nm thick analyte overlayer is also investigated. Figure 6(c) depicts the simulated transmission spectra of different refractive indices of analyte with the thickness fixed as 40 nm. Clearly, the resonance shows a red shift of 5.89 GHz with varying the refractive index from n = 1 to n = 4. In the inset of Fig. 6(c), the resonance frequency shift (FS) defined as fn − fn0, is calculated with different refractive indices of analytes, where fn is the resonance frequency with refractive index n and fn0 is the resonance frequency of n0 = 1. One can clearly observe that the FS increases linearly with refractive index increasing. For refractive index sensor, the frequency sensitivity is defined as the FS induced by the analyte index change [30]. With refractive indices increasing from 1.0 to 4.0, the sensitivity can be calculated as the slope of the linear fitting function of FS. By fitting the simulation data, the fitting function is expressed as FS = −2.06 + 1.96n, and the frequency sensitivity of 1.96 GHz per refractive index unit (RIU) can be obtained. For conventional THz-TDS measurements, a 200 ps scan time is required to achieve 5 GHz spectral resolution in the frequency domain and the sensitivity of 1.96 GHz/RIU requires a quite long scan time close to 1000 ps to discern the coated analyte response from the uncoated one [36]. However, it is highly desirable to have a fast process for the sensing procedure, thus needing to minimize the scan time as much as possible, which means the sensitivity of 1.96 GHz/RIU is a very low value and difficult to be used for sensing. In contrast, the sensitivity of the amplitude difference for the proposed metasurface is also investigated by varying the refractive index of 40 nm thick analyte overlayer. Figure 6(d) shows the transmission amplitude difference with changing the refractive index of analyte overlayer. With the refractive index changing from n = 1 to n = 2, the peak-to-peak amplitude difference |ΔT|, defined as |ΔTmax − ΔTmin|, can be calculated as 40.54%, where ΔTmax and ΔTmax is the maximum and minimum value in the amplitude difference. Moreover, in the inset of Fig. 6(d), the peak-to-peak amplitude difference |ΔT| with different refractive indices of analytes is calculated. Clearly, |ΔT| increases linearly with refractive index increasing and can be linearly fitted as |ΔT| = −0.28 + 0.32n. The corresponding sensitivity can be calculated as 0.32/RIU, which is higher than the value reported by Srivastava, et al. [37]. Therefore, for the nanofilm analyte with the same thickness, the traditional frequency shift sensing method with the sensitivity of 1.96 GHz/RIU is a very low value and difficult to be used for sensing, while the amplitude difference referencing technique has good sensing performance and more suitable for sensing nanofilm analytes at THz wavelength.
4. Conclusion
In summary, a TD-BIC inspired Fano resonance metasurface sensor composed of mirror symmetry broken SRRs was proposed and investigated, which enables sensing nanofilm analytes at THz frequencies. With the destructive interference between two anti-aligned magnetic dipoles and the calculated scattering power of TD, the physical mechanism of TD resonance was demonstrated. Meanwhile, with the asymmetry parameter decreasing, the transmission spectra changes from Fano to symmetry-protected BIC and the Q-factors obey the inverse square law, which demonstrates its BIC feature. For the first time to our knowledge, the TD-BIC inspired Fano resonance was utilized to effectively sense nanofilm analytes by using the amplitude difference referencing technique. Simulation results show the amplitude difference with a significant value can be easily observed comparing with the frequency shift under difference thicknesses of Ge overlayer. Moreover, with coating 40 nm analytes overlayer, the sensitivity of the amplitude difference can achieve 0.32/RIU, which also can be easily detected comparing with the frequency shift method with sensitivity of 1.96 GHz/RIU. Our proposed sensor combined with the amplitude difference referencing technique can be utilized to identify nanofilm analytes and pave a way for new generation THz label-free sensors.
Funding
National Natural Science Foundation of China (61675230, 61905276); China Postdoctoral Science Foundation (2018M643763, BX20180353); Natural Science Foundation of Shannxi Province (2020JQ-437); Open Research Fund of Key Laboratory of Spectral Imaging Technology, Chinese Academy of Sciences (LSIT201913N).
Disclosures
The authors declare no conflicts of interest.
References
1. I. B. Zel’Dovich, “Electromagnetic Interaction with Parity Violation,” J. Exp. Theor. Phys. 6(6), 1184 (1958).
2. A. Ceulemans, L. F. Chibotaru, and P. W. Fowler, “Molecular Anapole Moments,” Phys. Rev. Lett. 80(9), 1861–1864 (1998). [CrossRef]
3. V. M. Dubovik and V. V. Tugushev, “Toroid moments in electrodynamics and solid-state physics,” Phys. Rep. 187(4), 145–202 (1990). [CrossRef]
4. A. D. Boardman, K. Marinov, N. Zheludev, and V. A. Fedotov, “Dispersion properties of nonradiating configurations: Finite-difference time-domain modeling,” Phys. Rev. E 72(3), 036603 (2005). [CrossRef]
5. 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]
6. K. Marinov, A. D. Boardman, V. A. Fedotov, and N. Zheludev, “Toroidal metamaterial,” New J. Phys. 9(9), 324 (2007). [CrossRef]
7. 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]
8. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305(5685), 788–792 (2004). [CrossRef]
9. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz Magnetic Response from Artificial Materials,” Science 303(5663), 1494–1496 (2004). [CrossRef]
10. S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95(13), 137404 (2005). [CrossRef]
11. Z. Song, Y. Deng, Y. Zhou, and Z. Liu, “Terahertz toroidal metamaterial with tunable properties,” Opt. Express 27(4), 5792–5797 (2019). [CrossRef]
12. S. Han, M. Gupta, L. Cong, Y. K. Srivastava, and R. Singh, “Toroidal and magnetic Fano resonances in planar THz metamaterials,” J. Appl. Phys. 122(11), 113105 (2017). [CrossRef]
13. 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]
14. S. Lepeshov and Y. Kivshar, “Near-Field Coupling Effects in Mie-Resonant Photonic Structures and All-Dielectric Metasurfaces,” ACS Photonics 5(7), 2888–2894 (2018). [CrossRef]
15. 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]
16. G. Sun, L. Yuan, Y. Zhang, X. Zhang, and Y. Zhu, “Q-factor enhancement of Fano resonance in all-dielectric metasurfaces by modulating meta-atom interactions,” Sci. Rep. 7(1), 8128 (2017). [CrossRef]
17. K. Fan, I. V. Shadrivov, and W. J. Padilla, “Dynamic bound states in the continuum,” Optica 6(2), 169–173 (2019). [CrossRef]
18. C. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljacic, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]
19. H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32(6), 3231–3242 (1985). [CrossRef]
20. A. Krasnok, D. Baranov, H. Li, M.-A. Miri, F. Monticone, and A. Alú, “Anomalies in light scattering,” Adv. Opt. Photonics 11(4), 892–951 (2019). [CrossRef]
21. 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]
22. L. Cong and R. Singh, “Symmetry-Protected Dual Bound States in the Continuum in Metamaterials,” Adv. Opt. Mater. 7(13), 1900383 (2019). [CrossRef]
23. X. Chen and W. Fan, “Tunable Bound States in the Continuum in All-Dielectric Terahertz Metasurfaces,” Nanomaterials 10(4), 623 (2020). [CrossRef]
24. D. R. Abujetas, N. van Hoof, S. ter Huurne, J. Gómez Rivas, and J. A. Sánchez-Gil, “Spectral and temporal evidence of robust photonic bound states in the continuum on terahertz metasurfaces,” Optica 6(8), 996–1001 (2019). [CrossRef]
25. S. Han, L. Cong, Y. K. Srivastava, B. Qiang, M. V. Rybin, A. Kumar, R. Jain, W. X. Lim, V. G. Achanta, S. S. Prabhu, Q. J. Wang, Y. S. Kivshar, and R. Singh, “All-Dielectric Active Terahertz Photonics Driven by Bound States in the Continuum,” Adv. Mater. 31(37), 1901921 (2019). [CrossRef]
26. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017). [CrossRef]
27. A. Tittl, A. Leitis, M. Liu, F. Yesilkoy, D.-Y. Choi, D. N. Neshev, Y. S. Kivshar, and H. Altug, “Imaging-based molecular barcoding with pixelated dielectric metasurfaces,” Science 360(6393), 1105–1109 (2018). [CrossRef]
28. P. U. Jepsen, D. G. Cooke, and M. Koch, “Terahertz spectroscopy and imaging-Modern techniques and applications,” Laser Photonics Rev. 5(1), 124–166 (2011). [CrossRef]
29. W. Xu, L. Xie, and Y. Ying, “Mechanisms and applications of terahertz metamaterial sensing: a review,” Nanoscale 9(37), 13864–13878 (2017). [CrossRef]
30. R. Singh, W. Cao, I. Al-Naib, L. Cong, W. Withayachumnankul, and W. Zhang, “Ultrasensitive terahertz sensing with high-Q Fano resonances in metasurfaces,” Appl. Phys. Lett. 105(17), 171101 (2014). [CrossRef]
31. X. Chen, W. Fan, and C. Song, “Multiple plasmonic resonance excitations on graphene metamaterials for ultrasensitive terahertz sensing,” Carbon 133, 416–422 (2018). [CrossRef]
32. Y. K. Srivastava, L. Cong, and R. Singh, “Dual-surface flexible THz Fano metasensor,” Appl. Phys. Lett. 111(20), 201101 (2017). [CrossRef]
33. M. Gupta, Y. K. Srivastava, M. Manjappa, and R. Singh, “Sensing with toroidal metamaterial,” Appl. Phys. Lett. 110(12), 121108 (2017). [CrossRef]
34. I. Al-Naib, “Biomedical sensing with conductively coupled terahertz metamaterial resonators,” IEEE J. Sel. Top. Quantum Electron. 23(4), 1–5 (2017). [CrossRef]
35. H.-R. Park, X. Chen, N.-C. Nguyen, J. Peraire, and S.-H. Oh, “Nanogap-enhanced terahertz sensing of 1 nm thick (λ/106) dielectric films,” ACS Photonics 2(3), 417–424 (2015). [CrossRef]
36. I. Al-Naib, “Evaluation of amplitude difference referencing technique with terahertz metasurfaces for sub-micron analytes sensing,” J. King Saud Univ., Sci. 31(4), 1384–1387 (2019). [CrossRef]
37. Y. K. Srivastava, R. T. Ako, M. Gupta, M. Bhaskaran, S. Sriram, and R. Singh, “Terahertz sensing of 7 nm dielectric film with bound states in the continuum metasurfaces,” Appl. Phys. Lett. 115(15), 151105 (2019). [CrossRef]
38. F. D’Angelo, Z. Mics, M. Bonn, and D. Turchinovich, “Ultra-broadband THz time-domain spectroscopy of common polymers using THz air photonics,” Opt. Express 22(10), 12475–12485 (2014). [CrossRef]
39. M. F. Limonov, M. V. Rybin, A. N. Poddubny, and Y. S. Kivshar, “Fano resonances in photonics,” Nat. Photonics 11(9), 543–554 (2017). [CrossRef]
40. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5(1), 5753 (2014). [CrossRef]
41. M. Gupta, V. Savinov, N. Xu, L. Cong, G. Dayal, S. Wang, W. Zhang, N. I. Zheludev, and R. Singh, “Sharp Toroidal Resonances in Planar Terahertz Metasurfaces,” Adv. Mater. 28(37), 8206–8211 (2016). [CrossRef]
42. S. Han, L. Cong, F. Gao, R. Singh, and H. Yang, “Observation of Fano resonance and classical analog of electromagnetically induced transparency in toroidal metamaterials,” Ann. Phys. 528(5), 352–357 (2016). [CrossRef]
43. V. Savinov, V. Fedotov, and N. I. Zheludev, “Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials,” Phys. Rev. B 89(20), 205112 (2014). [CrossRef]
44. V. A. Fedotov, A. V. Rogacheva, V. Savinov, D. P. Tsai, and N. I. Zheludev, “Resonant Transparency and Non-Trivial Non-Radiating Excitations in Toroidal Metamaterials,” Sci. Rep. 3(1), 2967 (2013). [CrossRef]
45. T. C. Tan, E. Plum, and R. Singh, “Lattice-Enhanced Fano Resonances from Bound States in the Continuum Metasurfaces,” Adv. Opt. Mater. 8(6), 1901572 (2020). [CrossRef]
46. X. He, F. Liu, F. Lin, and W. Shi, “Investigation of terahertz all-dielectric metamaterials,” Opt. Express 27(10), 13831–13844 (2019). [CrossRef]
47. N. Vieweg, F. Rettich, A. Deninger, H. Roehle, R. Dietz, T. Göbel, and M. Schell, “Terahertz-time domain spectrometer with 90 dB peak dynamic range,” J. Infrared, Millimeter, Terahertz Waves 35(10), 823–832 (2014). [CrossRef]
