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Abstract
All-dielectric resonant metasurfaces are expected to boost Mie resonances with high Q-factors and enhanced electromagnetic fields owing to their large mode volumes and low material losses. However, the toroidal dipole (TD) and magnetic dipole (MD) are usually suppressed by other stronger multipoles due to their relatively weak coupling to the incident lights. In this work, the double resonances of TD and MD quasi-bound state in the continuum (quasi-BIC) are excited asymmetrically by breaking the geometric symmetry in an all-dielectric metasurface consisting of arrays of silicon I-bars and silicon Φ-disks, leading to their corresponding enhanced electric field confinements and high Q-factors. The sensing performances by these two resonances are investigated as well, achieving refractive index sensitivities of 784.8 nm/RIU and 630 nm/RIU, respectively. This work suggests a route to manipulate strong TD and MD quasi-BIC excitations and facilitates their practical applications such as nonlinear light sources and sensing.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Metasurfaces, as 2D forms of metamaterials, have extraordinary resonant optical responses and have been extensively employed in nanophotonics [1,2]. The surface plasmon resonances can be excited by sub-wavelength metallic metasurfaces, while their Q-factors are limited on account of high inherent resistive losses in the metals [3]. Therefore, all-dielectric metasurfaces with small dissipation and high laser damage threshold have been proposed, which can support Mie resonances with high Q-factors and low absorption losses [4]. They possess great capabilities of confining and enhancing the near field inside the nanoparticles, which govern the performance of many applications, such as harmonic generation [5,6], Raman scattering [7], and sensing [8,9].
Toroidal dipole (TD) is one of the members of Mie multipoles in all-dielectric metasurfaces, which originates from two inverse circulating currents [10–12]. Compared with electric dipole (ED) and magnetic dipole (MD), TD has weaker capabilities of confining and enhancing the near field [13,14]. Many researches about enhancing TD response have been reported. By manipulating the near-field coupling and the geometry arrangement, the strong and nearly pure TD resonance can be achieved in all-dielectric metasurfaces composed of clusters of subwavelength high-index dielectric cylinders [15,16]. Regulating the asymmetric permittivities of the components in all-dielectric metasurfaces is also an effective way to excite and boost the TD response [17], providing an efficient platform for sensing and optical nonlinearity. However, the electric field enhancements and Q-factors of TD resonances in the existing work are still insufficient, which require further explorations in structure design and physical mechanism innovation.
Bound state in the continuum (BIC) can produce extremely high Q resonances, which appears firstly in quantum mechanics, and then is reported in other fields such as optics, acoustics, hydrodynamics, and etc. [18–22]. A true BIC only appears in ideal lossless infinite structures or extreme values of parameters, and its resonance linewidth disappears while its Q-factor is infinite [23–25]. Normally, the BIC is symmetrically protected since the coupling constants vanish accidentally due to symmetry [26]. Recently it is revealed that symmetry-protected BIC can be transformed into quasi-BIC with a finite yet sharply high Q resonance by breaking the symmetry [27–29]. Some structures supporting quasi-BIC have also been reported, such as nanoholes [30,31], circular slots [32], and etc, implying outstanding performances in various applications, such as high Q cavity-mode laser [33], dynamical image tuning and display [34], and enhancing the nonlinear interactions between light and matter [35].
In this work, we propose the asymmetric excitations of TD resonance and MD quasi-BIC in an I-Φ shaped all-dielectric metasurface related to the parameter of distance difference. First, double resonances with giant enhancements of electromagnetic fields and extremely high Q-factors are excited. Then, by adjusting the distance difference between the I-bar and the neighboring Φ-disks, the physical mechanisms of double resonances are demonstrated. Finally, the sensing performances by these two resonances are investigated as well.
2. Structure design and double resonances properties
The proposed all-dielectric metasurface has a square-lattice pattern, as shown in Fig. 1, whose unit cell consists of a silicon I-shaped bar and a silicon Φ-shaped disk surrounded by air. The dispersive refractive indexes of silicon are taken from its experimentally measured data [36]. A y-polarized plane wave propagating along the z-axis is illuminated normally upon the metasurface. The numerical optical responses of the proposed silicon I-Φ shaped metasurface are simulated by commercial software COMSOL MultiphysicsTM based on the finite element method. Periodic boundary conditions are applied in the x- and y-directions, while perfectly matched layers are used to truncate the infinite space along the z-direction. The practical fabrication can be readily operated on silicon film by electron-beam lithography and reactive-ion etching techniques [37,38].
Fig. 1. Schematic illustration of the proposed silicon I-Φ shaped metasurface consisting of I-bars and Φ-disks. Geometric parameters: length of I-bars L = 790 nm, width and thickness of nanorods for both I-bars and Φ-disks W = 60 nm and T = 120 nm, inner and outer radius of Φ-disks Ri = 240 nm and Ro = 300 nm, and period of the metasurface P = 1000 nm. Distance difference Δg = g2 – g1.
As is well known, the bar-disk tandem structures have been proposed to support Fano resonances [37–5]. However, the excitations of the TD and MD resonances were usually difficult due to their relatively weak coupling to the incident lights. The design consideration of the proposed metasurface lies on the overlap between the anapole and the electric dipole (ED) resonances. The Φ-disk array supports structured anapole modes [39], while the I-bar array excites ED modes. When the geometric parameters are chosen carefully to let them overlap at a same wavelength of 1072 nm (spectrum not shown for simplicity), and the array is arranged by both of the I-bar and Φ-disk, the structured MD dark modes in the Φ-disks would be induced and suppress the structured anapole modes. This is because the dipole axis of the structured MD modes is out-of-plane, and it is easy to be excited by the magnetic field of a co-plane ED mode rather than that of the normally incident wave, which appears to be perpendicular to the axis [37].
To start with, we define Δg = g2 – g1 as the distance difference between the bar and the neighboring disks. As Δg ≠ 0, the symmetry of the structure will be broken. Figure 2(a) gives the transmission spectrum of the proposed metasurface for Δg = 200 nm. It can be seen that two prominently sharp dips, Resonance R1 and Resonance R2, appear on the almost transparent spectrum in the wavelength range of 1100 nm ∼ 1300 nm, indicating the double resonances are excited. The widths of both resonances are narrow, implying the high Q-factors and the strong electromagnetic field confinement. Indeed, one of the fascinating properties of the all-dielectric metasurface is its ability to confine the electric field inside the dielectric nanostructures, which facilitates the practical applications of nonlinear frequency conversion, nanolaser, and etc. Therefore, the average electric field enhancement factor is introduced to quantify the normalized electric field intensity averaged within the silicon nanostructure [40], which is expressed as followed
Fig. 2. Spectra as functions of wavelength. (a) Transmission, and (b) average enhancement factor |Eavg/Einc|2.
3. Results and discussions
3.1 Dependence on distance difference Δg
Figure 3(a) demonstrates the dependence of the transmission spectra on the distance difference Δg. It can be seen that the double resonances are symmetric with respect to Δg = 0. However, the excitations of these two resonances are asymmetric. With the decreasing of |Δg|, Resonance R1 has a red shift and the line width is broadened, while Resonance R2 has a blue shift and the line width is narrowed. Moreover, it is interesting that R2 disappears at Δg = 0, as indicated by the white dash circle. For both resonances, their Fano-like line shapes might vary dramatically, but their abilities to confine electromagnetic can be directly manifested on the spectra of average enhancement factor. We thus calculate the spectra of average enhancement factors and extract their corresponding properties.
Fig. 3. (a) Transmission spectra of the proposed metasurface as functions of wavelength and distance difference Δg. (b) and (c), dependences of average enhancement factor and Q-factor on Δg of Resonances R1 and R2, respectively. (d) Relationship between the Q-factor of R2 and the asymmetry parameter α.
First of all, the average enhancement factor extracted at the resonant wavelength of Resonance R1 and its corresponding Q-factor as the functions of Δg are given in Fig. 3(b). It can be seen that as |Δg| enlarges, the average enhancement factor is increased. The corresponding Q-factor is increased synchronously, indicating that the electric field localization comes inherently from the mode resonance. As |Δg| reaches 250 nm, the average enhancement factor and the Q-factor are as high as 3533 and 7885, which are 7 folds and 9 folds when compared to those of |Δg| = 0, respectively. Secondly, in contrast to R1, R2 experiences a totally different excitation, as shown in Fig. 3(c). The average enhancement factor and the Q-factor both grow exponentially and approach infinity as |Δg| decreases. As |Δg| reaches 25 nm, they are as high as 7917 and 17684, which are 34 folds and 62 folds when compared to those of |Δg| = 250 nm, respectively. These extremely high average enhancement factor and Q-factor indicate the great potentials of the proposed all-dielectric metasurface in the applications of nonlinear optics and nanolaser [41,42].
Mathematically, a true bound state in the continuum (BIC) has infinite value of the Q-factor, but it is in fact a “dark mode”, which does not manifest itself in the R2 transmission spectrum in Fig. 3(a). However, the average enhancement factor and Q-factor of R2 are not infinite but disappear suddenly as |Δg| decreases to be 0, and this can be regarded as the symmetry-protected BIC. Interestingly, the symmetry-protected BIC is easily destroyed by breaking the geometric symmetry to induce leakage, generating the quasi-BIC with radiative loss when |Δg| > 0 [43]. To further verify this judgment, we fit the relationship between the Q-factor of R2 and the asymmetry parameter α, as shown in Fig. 3(d). α is defined as Δg/g, and g is the distance between the I-bars and the Φ-disks at Δg = 0. It can be seen that the evolution of the Q-factor of R2 is consistent with the theoretical formula between the radiative Q-factor Qrad and the α [26],
The perfect match with the inverse quadratic law of Eq. (2) indicates that R2 can be recognized as a quasi-BIC, which is featured to have an extremely high Q-factor as α approaching zero.
In order to analyze the physical origins of these two resonances, the Cartesian multipole decompositions [44] are performed and those for the distance difference Δg = 0, 25 nm, 100 nm and 200 nm are presented in Fig. 4. For Resonance R1, although there are some other dipoles mixing in, the toroidal dipole (TD) contributes more. As Δg increases, the intensities of these dipoles change, but TD always dominates, and its line width is getting narrower. Hence, R1 can be theoretically attributed to be the TD. On the other hand, for R2, it is a pure MD resonance when Δg = 0. However, it is really weak and submerges on the flat curve of other dipole moments, as shown in Fig. 4(a). As the symmetry breaks and Δg increases to be 25 nm, this symmetry-protected BIC is transformed into the quasi-BIC, and the MD floats up and surpasses all other moments, as shown in Fig. 4(b). As Δg increases further, there are some other dipoles mixing in, and the line width of MD is getting wider. Nevertheless, the MD is the dominated dipole of R2, and its evolution corresponds to the Q-factors in Fig. 3(c).
Fig. 4. Cartesian multipole decompositions of the two resonances (R1 on the left, while R2 right) for (a) Δg = 0, (b) Δg = 25 nm, (c) Δg = 100 nm, and (d) Δg = 200 nm. ED(C), TD, MD, EQ, and MQ represent the Cartesian electric dipole, toroidal dipole, magnetic dipole, electric quadrupole and magnetic quadrupole, respectively.
To further verify the origins of these two resonances, Fig. 5 demonstrates the corresponding electromagnetic field distributions. For Resonance R1 in Fig. 5(a), when Δg = 0, there has two opposite circular currents, revealing the pattern of TD. However, these two currents evenly distribute on two sides of Φ-disk, which can be considered to be two MDs in opposite directions. Therefore, the destructive interference of these two opposite MDs leads to the weakest ability of confinement. When |Δg| > 0, as mentioned above, more dipoles are mixed in, and one of the circular currents is destroyed, leaving the other one. As Δg increases, the only existing circular current approaches to the center of Φ-disk, resulting in that the electric field is enhanced inside the dielectric structures, which corresponds to the dependence of average enhancement factor and Q-factor on Δg in Fig. 3(b). On the other hand, for the MD quasi-BIC of R2 in Fig. 5(b), when Δg = 0, the distributions of both electric and magnetic fields have no any feature. However, as Δg slightly increases to be 25 nm, the quasi-BIC is formed and the circular current of MD emerges around the center of Φ-disk, leading to the extremely high localization and enhancement of electric field inside the dielectric structures of Φ-disk. As Δg increases further, the circular current moves away from the center of Φ-disk, leading to the rapidly degradation of the electromagnetic field. Together with the evolution in Fig. 5(a), it is exactly the position of circular current with respect to the center of Φ-disk leading to the asymmetric excitations of R2 and R1, which have been demonstrated in Fig. 3(a)-(c).
Fig. 5. Normalized electric fields distributions |E/E0| on the xy-plane and normalized magnetic fields distributions |H/H0| on the xz-plane for different values of Δg. The electric field vectors (white arrows) and magnetic field vectors (black arrows) are also presented in their corresponding patterns, (a) R1 and (b) R2.
3.2 Refractive index sensing
Besides the ability to confine the electric field inside the dielectric nanostructure, the proposed metasurface also has the opportunity to be applied in refractive index (RI) sensing. In practical operations, the loss and the surface roughness of the metasurfaces will influence the resonances thus the sensing performance. Here, they are not considered for simplicity. Figures 6(a) and (b) give the resonant wavelength shifts as the function of the RI for Resonances R1 and R2, respectively. The sensing analyte is set as the infinite surrounding medium. It is interesting to be seen that the wavelength shifts of both resonances are proportional to the variation of RI no matter how Δg changes. This stems from the fact that the electric field energies of high Q resonances are mainly confined inside the dielectric nanostructure, and rarely distributed in the surrounding medium. Therefore, the RI sensing capabilities depend more on the modes themselves rather than on the electric field hot spot on the surrounding sensing medium [45,46].
Fig. 6. Dependences of resonant wavelength shift on refractive index of surrounding medium, (a) R1 and (b) R2.
Nevertheless, the structured modes supported by the proposed metasurface have high sensing abilities. The refractive index sensitivities (RISs) are as high as 784.8 nm/RIU and 630 nm/RIU for R1 and R2, respectively, and they are almost independent on the distance difference Δg. The zero wavelength shift for R2 under Δg = 0 is because there is no resonant mode formed due to the symmetry-protected BIC. The RISs based on all-dielectric metasurfaces have been previously explored in Table 1. Compared with these published works, the proposed metasurface has higher RISs. Together with the ease of fabrication resulting from the independence on Δg, the modes supported by our proposed metasurface have great potential for RI sensing application.
Table 1. Comparison of the sensitivity of similar works published previouslya
4. Conclusion
In summary, we have demonstrated that by breaking the geometric symmetry in an all-dielectric metasurface consisting of arrays of silicon I-bars and silicon Φ-disks, TD resonance and MD quasi-BIC are excited asymmetrically by the distance difference Δg, realizing giant enhancements of electromagnetic fields and extremely high Q-factors. Numerical results show that the average enhancement factor and the Q-factor of TD resonance are as high as 3533 and 7885 when Δg = 250 nm, and those of MD resonance governed by quasi-BIC can be as high as 7917 and 17684 when Δg = 25 nm. Furthermore, the sensing performances by these two resonances are investigated as well, achieving refractive index sensitivities of 784.8 nm/RIU and 630 nm/RIU, respectively, which are higher than those of previous works based on all-dielectric metasurfaces. Our proposed all-dielectric metasurface broadens the way to efficiently tailor strong TD and MD quasi-BIC excitations, and provides promising prospects for applications in light sources, sensing and many other aspects.
Funding
National Natural Science Foundation of China (11604276, 61871462); National Key Research and Development Program of China (2018YFC0603503).
Disclosures
The authors declare 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. A. E. Minovich, A. E. Miroshnichenko, A. Y. Bykov, T. V. Murzina, D. N. Neshev, and Y. S. Kivshar, “Functional and nonlinear optical metasurfaces,” Laser Photonics Rev. 9(2), 195–213 (2015). [CrossRef]
2. H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016). [CrossRef]
3. M. Allione, V. V. Temnov, Y. Fedutik, U. Woggon, and M. V. Artemyev, “Surface plasmon mediated interference phenomena in low-Q silver nanowire cavities,” Nano Lett. 8(1), 31–35 (2008). [CrossRef]
4. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]
5. H. Liu, C. Guo, G. Vampa, J. L. Zhang, T. Sarmiento, M. Xiao, P. H. Bucksbaum, J. Vučković, S. Fan, and D. A. Reis, “Enhanced high-harmonic generation from an all-dielectric metasurface,” Nat. Phys. 14(10), 1006–1010 (2018). [CrossRef]
6. W. Tong, C. Gong, X. Liu, S. Yuan, Q. Huang, J. Xia, and Y. Wang, “Enhanced third harmonic generation in a silicon metasurface using trapped mode,” Opt. Express 24(17), 19661–19670 (2016). [CrossRef]
7. S. Romano, G. Zito, S. Managò, G. Calafiore, E. Penzo, S. Cabrini, A. C. De Luca, and V. Mocella, “Surface-enhanced Raman and fluorescence spectroscopy with an all-dielectric metasurface,” J. Phys. Chem. C 122(34), 19738–19745 (2018). [CrossRef]
8. J. Zhang, W. Liu, Z. Zhu, X. Yuan, and S. Qin, “Strong field enhancement and light-matter interactions with all-dielectric metamaterials based on split bar resonators,” Opt. Express 22(25), 30889–30898 (2014). [CrossRef]
9. T. Ma, Q. Huang, H. He, Y. Zhao, X. Lin, and Y. Lu, “All-dielectric metamaterial analogue of electromagnetically induced transparency and its sensing application in terahertz range,” Opt. Express 27(12), 16624–16634 (2019). [CrossRef]
10. A. Sayanskiy, M. Danaeifar, P. Kapitanova, and A. E. Miroshnichenko, “All-dielectric metalattice with enhanced toroidal dipole response,” Adv. Opt. Mater. 6(19), 1800302 (2018). [CrossRef]
11. I. V. Stenishchev and A. A. Basharin, “Toroidal response in all-dielectric metamaterials based on water,” Sci. Rep. 7(1), 9468 (2017). [CrossRef]
12. 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]
13. A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B 82(4), 045404 (2010). [CrossRef]
14. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354(6314), aag2472 (2016).
15. 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]
16. 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]
17. 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]
18. H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32(6), 3231–3242 (1985). [CrossRef]
19. R. Parker, “Resonance effects in wake shedding from parallel plates: some experimental observations,” Journal of Sound and Vibration 4(1), 62–72 (1966). [CrossRef]
20. F. Ursell, “Trapping modes in the theory of surface waves,” Math. Proc. Cambridge Philos. Soc. 47(2), 347–358 (1951). [CrossRef]
21. D. Marinica, A. Borisov, and S. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]
22. E. N. Bulgakov and A. F. Sadreev, “Bound states in the continuum in photonic waveguides inspired by defects,” Phys. Rev. B 78(7), 075105 (2008). [CrossRef]
23. R. F. Ndangali and S. V. Shabanov, “Electromagnetic bound states in the radiation continuum for periodic double arrays of subwavelength dielectric cylinders,” J. Math. Phys. 51(10), 102901 (2010). [CrossRef]
24. C. W. Hsu, B. Zhen, J. Lee, S. -L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljačić, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]
25. C. W. Hsu, B. Zhen, A. D. Stone, J. D. Joannopoulos, and M. Soljačić, “Bound states in the continuum,” Nat. Rev. Mater. 1(9), 16048 (2016). [CrossRef]
26. 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]
27. Z. F. Sadrieva, I. S. Sinev, K. L. Koshelev, A. Samusev, I. V. Iorsh, O. Takayama, R. Malureanu, A. A. Bogdanov, and A. V. Lavrinenko, “Transition from optical bound states in the continuum to leaky resonances: Role of substrate and roughness,” ACS Photonics 4(4), 723–727 (2017). [CrossRef]
28. M. A. Belyakov, M. A. Balezin, Z. F. Sadrieva, P. V. Kapitanova, E. A. Nenasheva, A. F. Sadreev, and A. A. Bogdanov, “Experimental observation of symmetry protected bound state in the continuum in a chain of dielectric disks,” Phys. Rev. A 99(5), 053804 (2019). [CrossRef]
29. S. Li, C. Zhou, T. Liu, and S. Xiao, “Symmetry-protected bound states in the continuum supported by all-dielectric metasurfaces,” Phys. Rev. A 100(6), 063803 (2019). [CrossRef]
30. S. Xie, C. Xie, S. Xie, J. Zhan, Z. Li, Q. Liu, and G. Tian, “Bound states in the continuum in double-hole array perforated in a layer of photonic crystal slab,” Appl. Phys. Express 12(12), 125002 (2019). [CrossRef]
31. S. Xie, S. Xie, J. Zhan, C. Xie, G. Tian, Z. Li, and Q. Liu, “Bound states in the continuum in a T-shape nanohole array perforated in a photonic crystal slab,” Plasmonics 15(5), 1261–1271 (2020). [CrossRef]
32. J. Algorri, F. Dell’Olio, P. Roldán-Varona, L. Rodríguez-Cobo, J. López-Higuera, J. Sánchez-Pena, and D. Zografopoulos, “Strongly resonant silicon slot metasurfaces with symmetry-protected bound states in the continuum,” Opt. Express 29(7), 10374–10385 (2021). [CrossRef]
33. 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]
34. L. Xu, K. Zangeneh Kamali, L. Huang, M. Rahmani, A. Smirnov, R. Camacho-Morales, Y. Ma, G. Zhang, M. Woolley, and D. Neshev, “Dynamic nonlinear image tuning through magnetic dipole quasi-BIC ultrathin resonators,” Adv. Sci. 6(15), 1802119 (2019). [CrossRef]
35. K. Koshelev, Y. Tang, K. Li, D.-Y. Choi, G. Li, and Y. Kivshar, “Nonlinear metasurfaces governed by bound states in the continuum,” ACS Photonics 6(7), 1639–1644 (2019). [CrossRef]
36. M. R. Shcherbakov, P. Vabishchevich, A. Shorokhov, K. E. Chong, D. Y. Choi, I. Staude, A. E. Miroshnichenko, D. N. Neshev, A. A. Fedyanin, and Y. S. Kivshar, “Ultrafast all-optical switching with magnetic resonances in nonlinear dielectric nanostructures,” Nano Lett. 15(10), 6985–6990 (2015). [CrossRef]
37. 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]
38. Y. Yang, W. Wang, A. Boulesbaa, I. I. Kravchenko, D. P. Briggs, A. Puretzky, D. Geohegan, and J. Valentine, “Nonlinear Fano-resonant dielectric metasurfaces,” Nano Lett. 15(11), 7388–7393 (2015). [CrossRef]
39. W. Wang, X. Zhao, L. Xiong, L. Zheng, Y. Shi, Y. Liu, and J. Qi, “Broken symmetry theta-shaped dielectric arrays for a high Q-factor Fano resonance with anapole excitation and magnetic field tunability,” OSA Continuum 2(2), 507–517 (2019). [CrossRef]
40. J. Yao, Y. Wu, J. Liu, N. Liu, and Q. H. Liu, “Enhanced optical bistability by coupling effects in magnetic metamaterials,” J. Lightwave Technol. 37(23), 5814–5820 (2019). [CrossRef]
41. Y. Gao, Y. Fan, Y. Wang, W. Yang, Q. Song, and S. Xiao, “Nonlinear holographic all-dielectric metasurfaces,” Nano Lett. 18(12), 8054–8061 (2018). [CrossRef]
42. W. Bi, X. Zhang, M. Yan, L. Zhao, T. Ning, and Y. Huo, “Low-threshold and controllable nanolaser based on quasi-BIC supported by an all-dielectric eccentric nanoring structure,” Opt. Express 29(8), 12634–12643 (2021). [CrossRef]
43. Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, “High-Q quasi-bound states in the continuum for nonlinear metasurfaces,” Phys. Rev. Lett. 123(25), 253901 (2019). [CrossRef]
44. 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, and N. I. Zheludev, “Optical anapole metamaterial,” ACS Nano 12(2), 1920–1927 (2018). [CrossRef]
45. N. Liu, M. Mesch, T. Weiss, M. Hentschel, and H. Giessen, “Infrared perfect absorber and its application as plasmonic sensor,” Nano Lett. 10(7), 2342–2348 (2010). [CrossRef]
46. G. Cai, W. Li, Y. Chen, N. Liu, Z. Song, and Q. H. Liu, “Modeling and design of a plasmonic sensor for high sensing performance and clear registration,” IEEE Photon. J. 8(1), 4801011 (2016).
47. Y. Zhang, W. Liu, Z. Li, Z. Li, H. Cheng, S. Chen, and J. Tian, “High-quality-factor multiple Fano resonances for refractive index sensing,” Opt. Lett. 43(8), 1842–1845 (2018). [CrossRef]
48. D. N. Maksimov, V. S. Gerasimov, S. Romano, and S. P. Polyutov, “Refractive index sensing with optical bound states in the continuum,” Opt. Express 28(26), 38907–38916 (2020). [CrossRef]
49. W. Su, Y. Ding, Y. Luo, and Y. Liu, “A high figure of merit refractive index sensor based on Fano resonance in all-dielectric metasurface,” Results Phys. 16, 102833 (2020). [CrossRef]
