Open Access
Add to BibSonomy
Abstract
We investigate the bound states in the continuum (BICs) in dielectric metasurfaces consisting of asymmetric dual rectangular patches in the unit cell of a square lattice. Various types of BICs are identified in the metasurface at normal incidence, associated with very large quality factors and vanishing spectral linewidths. In particular, symmetry-protected (SP) BICs occur when the four patches are fully symmetric, which exhibit antisymmetric field patterns that are decoupled from the symmetric incident waves. By breaking the symmetry of patch geometry, the SP BICs degrade to quasi-BICs that are characterized by Fano resonance. Accidental BICs and Friedrich-Wintgen (FW) BICs occur when the asymmetry is introduced in the upper two patches, while holding the lower two patches symmetric. The accidental BICs occur on isolated bands when the linewidth of either the quadrupole-like mode or LC-like mode vanishes by tuning the upper vertical gap width. The FW BICs appear when the avoided crossing is formed between the dispersion bands of dipole-like and quadrupole-like modes by tuning the lower vertical gap width. At a special asymmetry ratio, the accidental BICs and FW BICs may appear in the same transmittance or dispersion diagram, accompanied with the concurrence of dipole-like, quadrupole-like, and LC-like modes.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Bound states in the continuum (BICs) are non-leaky localized resonance modes that coexist with a continuous spectrum of radiating waves [1–3]. They are non-radiative waves and possess extremely large quality factors with vanishing spectral linewidths. As a concept introduced in quantum mechanics by von Neumann and Wigner in 1929 [4], the existence of BICs was explained by Friedrich and Wintgen in 1985 as the result of destructive interference between two resonances coupled to the same radiation channel [5]. Since the initial proposal of BICs in quantum systems, BICs have been identified in different systems as photonic, acoustic, water waves, and mathematical [6–12]. The unique properties of BICs have led to various applications [13], including lasers [14], sensors [15,16], filters [17], low-loss fibers [18,19], and nonlinear optics [20–23].
Metasurfaces are a kind of artificial sheet materials with subwavelength-scaled patterns in the horizontal directions. The metasurfaces with large quality factors, such as those of BICs, provide an efficient platform for ultrasensitive sensors, enhanced light emission, perfect absorbers, and nonlinear optics with strong localization of electromagnetic energy [24–26]. However, a true BIC is the mathematical abstraction and its realization demands infinite size of the structure or having zero or infinite permittivity [27,28]. It is therefore not possible for the BIC to couple with the external excitation without making any physical changes to the metasurfaces. In order to identify the BICs from the incident radiating fields, some modifications on the metasurfaces are necessary, for instance, to introduce structural perturbations such that the BICs can be coupled with the external excitation and partially leak out to the far field [29].
In general, BICs originate from two physical mechanisms. Symmetry-protected (SP) BICs are created from the first mechanism, that is, symmetry mismatch. When a trapped mode with a given symmetry is embedded into the continuous frequency spectrum with a distinct symmetry, the leakage of the trapped mode is forbidden [30,31], leading to the formation of SP BICs. This type of BICs have been identified in the metasurfaces composed of arrays of meta-atoms such as tilted elliptic strip or rectangular bar pairs [32–35], nanodisks with holes [36], circular split rings [37,38], rectangular blocks [39–41], rectangular strip pairs [42–47], and square split rings [48–51]. Ideal BICs have infinite quality factors and zero spectral linewidths [20,27], while in most real structures they are degraded to quasi-BICs with finite (though very large) quality factors and nonzero (though very small) spectral linewidths [52,53]. Quasi-BICs usually exist in the structures that support SP BICs by breaking the symmetry in geometry, such as altering the orientation angle [32–35], adding or removing part [37–48], or deviating the shape to break the $C_2$ symmetry [36,49,51]. Quasi-BICs also exist in the structure by incorporating an ohmic loss in the constituent material, which reduces somewhat the quality factor [50].
Accidental (or single-resonance parametric) BICs and Friedrich–Wintgen (FW) BIC are created from the second mechanism, that is, wave interference. An accidental BIC is usually located on an isolated dispersion band, where a single resonance may evolve ’accidentally’ into a BIC when enough parameters are tuned. This feature can be thought of as the cancellation of radiations from two (or more) constituting waves through parameter tuning. On the other hand, a FW BIC is generally found in the vicinity of avoided crossing between two dispersion bands, which arises because of the destructive interference of two guided mode resonances coupled to the same radiation channel [54]. In practice, the accidental BICs can be formed by deviating the cross-shaped structure [55], and the FW BICs can be created by tuning the geometric parameters of the split-ring structures [56–59].
In the present study, we investigate the BICs in dielectric metasurfaces consisting of asymmetric dual rectangular patches in the unit cell of a square lattice. Various types of BICs are identified in the metasurface at normal incidence, including SP BICs, accidental BICs, and FW BICs. The formation of BICs are explained by the dipole-like, quadrupole-like, and LC-like modes, with the illustration of out-of-plane magnetic field and in-plane displacement current patterns. In particular, SP BICs occur when the four patches are fully symmetric, which exhibit antisymmetric field distributions that are decoupled from the symmetric incident waves. An asymmetry parameter is introduced for the upper two patches, while holding the lower two patches symmetric. By breaking the symmetry in geometry, SP BICs are degraded to quasi-BICs that are characterized by Fano resonance. In particular, the accidental BIC occurs on an isolated band of either the quadrupole-like or LC-like mode when the linewidth vanishes ’accidentally’ and a very large quality factor is attained at its peak by tuning the upper vertical gap width. On the other hand, the FW BIC occurs near the avoided crossing between the dispersion bands of dipole-like and quadrupole-like modes, in which the linewidth of one of the two modes vanishes and a very large quality factor is attained at its peak by tuning the lower vertical gap width. At a special asymmetry ratio, the accidental BICs and FW BICs may appear in the same transmittance or dispersion diagram, accompanied with the concurrence of dipole-like, quadrupole-like, and LC-like modes.
2. Theory and model design
Consider a dielectric metasurface consisting of asymmetric dual patches in the unit cell of a square lattice, as schematically shown in Fig. 1. Four rectangular dielectric patches of thickness $h$ are arranged inside a square of side length $l$ in the unit cell of period $a$, which is deposited on a thick dielectric substrate with a much larger thickness than that of the patch. A horizontal gap of width $g_0$ is between the upper and lower two patches, while a vertical gap of width $g_1$ ($g_2$) is between the upper (lower) two patches. In the present work, an asymmetry parameter is defined for the upper two patches as $\alpha \equiv {\delta }/{\delta _{\max }}$ [56], where $\delta$ is the distance from the center of upper vertical gap to the symmetry line (vertical line along the center of unit cell) and $\delta _{\max }$ is the maximum of $\delta$, where the upper right patch width is equal to the default gap width (50 nm). The lower vertical gap is located at the center of the unit cell. In the present study, $a$ = 800 nm, $l$ = 720 nm, $h$ = 40 nm, and $g_0$ = 50 nm are used as the basic geometric parameters for the metasurface. Note here that $\alpha =0\%$ ($\delta =0$ nm) for the symmetric case, and $\alpha =100\%$ ($\delta =720/2-50-50/2=285$ nm) for the most asymmetric case. In order to have a large contrast in symmetry between the upper two and the lower two patches, which is beneficial to the formation of BICs in the present structure, the lower patches remain symmetric through the study.
Fig. 1. Schematic diagram of the unit cell for the asymmetric dual-patch dielectric metasurface deposited on a thick dielectric substrate.
To analyze the properties of BICs, we calculate the transmittance (ratio of the transmitted to incident power) based on the finite element method, which has been employed to study the transmission characteristics of surface structures with patches or apertures [60,61]. A semi-infinite glass substrate with the refractive index 1.5 is assumed in the calculations, with periodic boundary conditions applied at the unit cell boundary and perfectly matched layers on top and bottom of the computational domain [62]. In the preset study, the dielectric patches are assumed to have the refractive index 5, which is valid in the spectral region of interest [63–66]. A relatively high dielectric contrast between the patches and background or substrate causes a large portion of electromagnetic fields confined inside or near the structure, which is advantageous to identify the BICs in the metasurfaces. In the present study, the incident wave is normal to the metasurface and polarized parallel (perpendicular) to the horizontal (vertical) gap. This polarization gives a more significant effect (than the other polarization) of the asymmetry on the resonance modes in the present configuration.
Meanwhile, to identify various types of the BICs, we compute the eigenfrequencies based on the plane wave expansion method and the supercell approach [67–70]. An important factor for evaluating BICs is the quality factor ($Q$ factor) based on the eigenfrequency $\Lambda$, which is defined as $Q=\textrm{Re}[\Lambda ]/\left (2\textrm{Im}[\Lambda ]\right )$ [71], where $\textrm{Re}[\cdot ]$ and $\textrm{Im}[\cdot ]$ denote the real and imaginary parts, respectively. Another definition of the quality factor is based on the spectral response as $Q=f_r/\Delta f_r$, where $f_r$ is the resonance frequency and $\Delta f_r$ is the linewidth (full width at half maximum) of the transmittance (or reflectance). The above two definitions of the quality factor are nearly equivalent and amount to the ratio of energy stored in the oscillating resonator to the energy dissipated per cycle by damping processes [72]. A larger quality factor corresponds to a narrower linewidth with a smaller damping rate and a longer lifetime for the resonator. Theoretically, the linewidth of an ideal BIC becomes zero and the quality factor tends to infinity [20,27].
3. Results and discussion
3.1 Resonance modes for the formation of BICs
Resonance modes are usually leaky when they are located in the continuous frequency spectrum [27]. BICs are a special type of resonance modes that are non-leaky, which may occur when the structure is carefully designed with enough parameters to be tuned. A few representative resonance modes are pertaining to the formation of BICs in the metasurfaces. Figure 2(a) shows the transmittance curve of the asymmetric dual-patch dielectric metasurface as a function of the wavelength for maximum asymmetry ($\alpha = 100\%$) and default vertical gap widths ($g_1=g_2=50$ nm). Three types of resonance modes are observed in the transmittance curve, which are Fano resonance in nature, that is, a transmission peak is accompanied nearby with a transmission dip. These resonance modes are closely related to the formation of BICs in the present structure, as will be discussed in detail in the following subsections. In Figs. 2(b)–2(e), the features of resonance modes at the transmittance peaks are illustrated with the patterns of out-of-plane magnetic fields (Re[$H_z$]) and in-plane displacement currents (Re[$\partial \mathbf{D}/\partial t$]). In particular, the patterns in Figs. 2(b) and 2(c) correspond to two dipole-like modes, which are labeled by D$_1$ and D$_2$, and the patterns in Figs. 2(d) and 2(e) correspond to the quadrupole-like mode labeled by Q and the LC-like mode labeled by LC, respectively.
Fig. 2. Resonance modes for the formation of BICs in the asymmetric dual-patch dielectric metasurface. (a) Transmittance curve as a function of the wavelength for $\alpha = 100\%$ and $g_1=g_2=50$ nm. Out-of-plane magnetic fields (Re[$H_z$]) in color and in-plane displacement currents (Re[$\partial \mathbf{D}/\partial t$]) by arrows for (b), (c) dipole-like modes [labeled by D$_1$, D$_2$ in (a)], (d) quadrupole-like mode [labeled by Q in (a)], and (e) LC-like mode [labeled by LC in (a)]. Field intensities are normalized to have a unity maximum.
Notice that the dipole-like and quadrupole-like modes exhibit one pair and two pairs, respectively, of dominate field regions with positive and negative values. The dipole-like modes are ubiquitous in a variety of structures when the resonance occurs, which may be related to the formation of accidental BICs, whereas the quadrupole-like modes are essential to the formation of SP BICs and have been observed in square split-ring [56,73] and cross-shaped [55] structures. In the present study, the dipole-like and quadrupole-like modes are further two relevant modes for the formation of FW BICs. On the other hand, the LC-like mode refers to the resonance with circulating displacement currents on the unit cell, which mimics the LC circuit consisting of an inductor and a capacitor connected in series. This mode can be effectively regarded as a magnetic dipole and has been identified in split-ring structures [56,57,73], which is related to the formation of accidental BICs in the present study. Note here that the field and current patterns of the resonance modes at the transmittance dips are basically similar to, but slightly different from, those at the transmittance peaks. The interference effects in the former and the latter are either constructive or destructive in the neighborhood of the resonance.
3.2 Symmetry-protected BICs
Figure 3(a) shows the transmittance diagram of the asymmetric dual-patch dielectric metasurface as a function of the asymmetry parameter $\alpha$ and the wavelength for default vertical gap widths ($g_1=g_2=50$ nm). The dispersion bands (blue and purple dashed lines) are overlaid in the same diagram to show the consistent trends with the transmittance. There are two dispersion bands identified as the quadrupole-like and LC-like modes, whose wavelengths increase with the asymmetry. The transmittance curves as a function of the wavelength for selected $\alpha$ are plot in Fig. 3(b). For $\alpha =0$, the metasurface is composed of four identical square patches and the system holds the $C_4$ (four-fold cyclic) symmetry. A BIC occurs on the dispersion band of quadrupole-like mode, which is identified as the SP BIC [20,41,47]. For this BIC, the incident field is decoupled from the resonance mode because of the symmetry mismatch. Equivalently, the resonance mode will not radiate to the far field and becomes a bound state without radiative leakage.
Fig. 3. Symmetry-protected BICs in the asymmetric dual-patch dielectric metasurface. (a) Transmittance diagram as a function of the asymmetry parameter and the wavelength for $g_1=g_2=50$ nm. (b) Transmittance curves as functions of the wavelength for selected asymmetry parameters. (c) Quality factor as a function of the asymmetry parameter. Black line is a fitted curve of the inverse-square law between $Q$ and $\alpha$. Gray region is the range for $\alpha >0.2$. Inset is the field pattern of the SP BIC.
Note that the transmittance curve becomes flat near the SP BIC, indicating that the corresponding resonance mode is not interacting with the incident field. The SP BICs usually, though not always, occur in symmetric structures when the system holds symmetry in both the geometry and incidence. On a special occasion, the SP BIC may appear in an asymmetric structure when a certain azimuthal angle of polarization (about the incident direction) is introduced [56]. For $\alpha \ne 0$, the vertical gap between the upper two patches is shifted to the right from the symmetry line and the $C_4$ symmetry is broken (cf. Figure 1). Two Fano resonances emerge on the transmittance curve, one of the quadrupole-like mode and the other of the LC-like mode [cf. Figure 3(b)]. The former originates from the structural asymmetry, whereas the latter is excited when the incident electric field is polarized in the direction normal to the vertical gap [73].
SP BICs occur when the structure has a perfect symmetry in geometry [56]. By introducing a certain asymmetry in the structure (or a small oblique incidence angle), the original SP BIC will somehow couple to the incident field, and the ideal BIC degrades to a quasi-BIC with finite quality factor and nonzero linewidth. In this situation, quasi-BICs are represented in the form of Fano resonance [37,49,73–77], which are rather significant with broadened linewidths as $\alpha$ increases, showing a strong dependence of the quasi-BICs on the asymmetry parameter. In particular, the quality factor $Q$ of the quasi-BIC as a function of $\alpha$ is fitted to the inverse-square law $Q\propto \alpha ^{-2}$ for $\alpha <0.2$ [20,55], as shown in Fig. 3(c). In this range, the inverse-square law serves as a useful empirical rule for evaluating the quality factors of the quasi-BICs. For $\alpha =0$, the structure has a perfect symmetry and the quality factor is extremely large (infinity in theory). The corresponding resonance field pattern, as shown in the inset, is an ideal quadrupole mode. For $\alpha >0.2$ (gray region), the quality factor deviates from the inverse-square law with the discrepancy increasing with the asymmetry parameter. In this range, the inverse-square law is not exactly valid.
3.3 Accidental BICs
Figure 4(a) shows the transmittance diagram of the asymmetric dual-patch dielectric metasurface as a function of the upper vertical gap width $g_1$ and the wavelength for maximum asymmetry ($\alpha =100\%$) and default lower vertical gap width ($g_2=50$ nm). The dispersion bands (blue and purple dashed lines) are overlaid in the same diagram to show consistent trends with the transmittance. There are two dispersion bands identified as the quadrupole-like and LC-like modes, whose wavelengths decrease with the gap width. In Fig. 4(b), the transmittance curves as a function of the wavelength are plot for selected gap widths. Note that the linewidth of quadrupole-like or LC-like mode vanishes ’accidentally’ at a certain gap width ($g_1=175$ nm or 105 nm, respectively), where the transmittance curve becomes flat and a very large quality factor is attained at its peak, as shown in Fig. 4(c). This is the characteristic of accidental BICs [55,78] that emerge on isolated bands because of the destructive interference between two or more leakage channels. In the present study, the accidental BIC can be either the quadrupole-like or LC-like mode that occurs when the upper vertical gap width is tuned and the lower gap width remains unchanged. Compared to the quadrupole-like mode, the LC-like mode occurs at a smaller gap width, where the circulating displacement currents are more enhanced. The accidental BICs also occur for $0<\alpha <100\%$ at different gap widths.
Fig. 4. Accidental BICs in the asymmetric dual-patch dielectric metasurface. (a) Transmittance diagram as a function of the upper vertical gap width $g_1$ and the wavelength for $\alpha =100\%$ and $g_2=50$ nm. (b) Transmittance curves as functions of the wavelength for selected gap widths. (c) Quality factors as functions of the gap width $g_1$. Positions of the accidental BICs are marked by triangles.
3.4 Friedrich-Wintgen BICs
Figure 5(a) shows the transmittance diagram of the asymmetric dual-patch dielectric metasurface as a function of the lower vertical gap width $g_2$ and the wavelength for maximum asymmetry ($\alpha =100\%$) and default upper vertical gap width ($g_1=50$ nm). The dispersion bands (yellow and blue dashed lines) are overlaid in the same diagram to show consistent trends with the transmittance. There are two dispersion bands identified as the dipole-like and quadrupole-like modes, whose wavelengths decrease with the gap width. In particular, they approach other at a certain gap width ($g_2=322$ nm) near which the avoided crossing [5,54,55] is formed between the two bands. In Fig. 5(b), the transmittance curves as a function of the wavelength are plot for selected gap widths. The avoided crossing point corresponds to the vanishing linewidth of the dipole-like mode and a very large quality factors is attained at its peak, as shown in Fig. 5(c). Meanwhile, the quality factor of the quadrupole-like mode is much smaller. The above feature is characteristic of the FW BICs [55,56] formed by tuning the interaction between two or more resonances.
Fig. 5. Friedrich-Wintgen BICs in the asymmetric dual-patch dielectric metasurface. (a) Transmittance diagram as a function of the lower vertical gap width $g_2$ and the wavelength for $\alpha =100\%$ and $g_1=50$ nm. (b) Transmittance curves as functions of the wavelength for selected gap widths. (c) Quality factors as functions of the gap width $g_2$. Position of the FW BIC is marked by circle.
The property of FW BICs can be understood from the coupled mode theory for a simple system with two resonances: $i{\partial A}/{\partial t}={\cal H}A$ [1], where $A=\left (A_1,A_2\right )^T$ are the amplitudes of resonances and
In the present configuration, the dispersion bands of dipole-like and quadrupole-like modes approach each other when the lower vertical gap width is tuned and the upper vertical gap width remains unchanged. The interference in the two resonance modes causes an avoided crossing between the dispersion bands when a like symmetry exists in the field distribution of the two modes. This is required when two dispersion bands repel each other near the avoided crossing point, and is attributed to some common mode symmetry that sustains in the asymmetric structure [56]. This feature is manifest on the dipole modes, quadrupole modes, and LC modes, as illustrated in Figs. 2(b)–2(e). Under suitable circumstances, the linewidth of one resonance may vanish when a specific value of the system parameter fulfills the condition of Eq. (2) in the tuning process, leading to the formation of a FW BIC. The FW BICs also occur for $0<\alpha <100\%$ at different gap widths.
Finally, the accidental BICs and FW BICs stated in previous two subsections can be identified in the same transmittance diagram at a special asymmetry ratio ($\alpha =40\%$) and default lower vertical gap width ($g_2=50$ nm), as shown in Fig. 6(a). In this case, the dispersion bands of dipole-like, quadrupole-like, and LC-like modes (yellow, blue, and purple dashed lines, respectively) are located in the vicinity of each other. By tuning the upper vertical gap width $g_1$, the accidental BIC emerges on the dispersion band of LC-like mode at a certain gap width ($g_1=58$ nm), where the linewidth on the transmittance curve vanishes [cf. Figure 6(b)] and a very large quality factor is attained at its peak value [cf. Figure 6(c)]. On the other hand, the FW BIC occurs near the avoided crossing between the dipole-like and quadrupole-like modes at a certain gap width ($g_1=74$ nm), where the linewidth of quadrupole-like mode vanishes [cf. Figure 6(b)] and a very large quality factor is attained at its peak, while the quality factor of the dipole-like mode is much smaller [cf. Figure 6(c)]. In this configuration, the lower two patches are symmetric and remain unchanged. The dispersion of dipole modes is less sensitive to the change of upper vertical gap width.
Fig. 6. Accidental BICs and Friedrich-Wintgen BICs in the asymmetric dual-patch dielectric metasurface. (a) Transmittance diagram as a function of the lower vertical gap width $g_1$ and the wavelength for $\alpha =40\%$ and $g_2=50$ nm. (b) Transmittance curves as functions of the wavelength for selected gap widths. (c) Quality factors as functions of the gap width $g_1$. Positions of the accidental BIC and FW BIC are marked by triangle and circle, respectively.
4. Concluding remarks
In conclusion, we have investigated the BICs in asymmetric dual-patch dielectric metasurfaces. Typical features of BICs as very large quality factors and vanishing spectral linewidths are identified in the underlying structure based on the transmittance diagram and dispersion bands. In particular, SP BICs occur on the dispersion bands of quadrupole-like modes with antisymmetric field patterns. Accidental BICs emerge on isolated bands of either the dipole-like or LC-like modes by tuning the upper vertical gap width. FW BICs appear when the avoided crossing occurs between the dipole-like and quadrupole-like modes by tuning the lower vertical gap width. At a special asymmetry ratio, both the accidental BICs and FW BICs can be observed in the same transmittance or dispersion diagram, accompanied with the concurrence of dipole-like, quadrupole-like, and LC-like modes.
Funding
National Science and Technology Council (MOST 111-2221-E-002-068-MY3).
Acknowledgments
This work was supported in part by Ministry of Science and Technology, Taiwan (MOST 111-2221-E-002-068-MY3).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
References
1. C. W. 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]
2. C. Peng, “Trapping light in the continuum-from fantasy to reality,” Sci. Bull. 65(18), 1527–1532 (2020). [CrossRef]
3. K. Koshelev, A. Bogdanov, and Y. Kivshar, “Engineering with bound states in the continuum,” Opt. Photonics News 31(1), 38–45 (2020). [CrossRef]
4. J. Von Neumann and E. Wigner, “Uber merkwürdige diskrete eigenwerte phys,” Z 30, 465–467 (1929).
5. H. Friedrich and D. Wintgen, “Interfering resonances and bound states in the continuum,” Phys. Rev. A 32(6), 3231–3242 (1985). [CrossRef]
6. M. Watts, S. Johnson, H. Haus, and J. Joannopoulos, “Electromagnetic cavity with arbitrary q and small modal volume without a complete photonic bandgap,” Opt. Lett. 27(20), 1785–1787 (2002). [CrossRef]
7. D. Marinica, A. Borisov, and S. Shabanov, “Bound states in the continuum in photonics,” Phys. Rev. Lett. 100(18), 183902 (2008). [CrossRef]
8. M. I. Molina, A. E. Miroshnichenko, and Y. S. Kivshar, “Surface bound states in the continuum,” Phys. Rev. Lett. 108(7), 070401 (2012). [CrossRef]
9. C. W. Hsu, B. Zhen, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Bloch surface eigenstates within the radiation continuum,” Light: Sci. Appl. 2(7), e84 (2013). [CrossRef]
10. R. Porter and D. Evans, “Embedded rayleigh-bloch surface waves along periodic rectangular arrays,” Wave Motion 43(1), 29–50 (2005). [CrossRef]
11. C. Linton and P. McIver, “Embedded trapped modes in water waves and acoustics,” Wave Motion 45(1-2), 16–29 (2007). [CrossRef]
12. H. Kruger, “On the existence of embedded eigenvalues,” J. Math. Anal. Appl. 395(2), 776–787 (2012). [CrossRef]
13. S. I. Azzam and A. V. Kildishev, “Photonic bound states in the continuum: from basics to applications,” Adv. Opt. Mater. 9(1), 2001469 (2021). [CrossRef]
14. A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kante, “Lasing action from photonic bound states in continuum,” Nature 541(7636), 196–199 (2017). [CrossRef]
15. B. Zhen, S.-L. Chua, J. Lee, A. W. Rodriguez, X. Liang, S. G. Johnson, J. D. Joannopoulos, M. Soljacic, and O. Shapira, “Enabling enhanced emission and low-threshold lasing of organic molecules using special Fano resonances of macroscopic photonic crystals,” Proc. Natl. Acad. Sci. U. S. A. 110(34), 13711–13716 (2013). [CrossRef]
16. Y. Liu, W. Zhou, and Y. Sun, “Optical refractive index sensing based on high-Q bound states in the continuum in free-space coupled photonic crystal slabs,” Sensors 17(8), 1861 (2017). [CrossRef]
17. J. M. Foley, S. M. Young, and J. D. Phillips, “Symmetry-protected mode coupling near normal incidence for narrow-band transmission filtering in a dielectric grating,” Phys. Rev. B 89(16), 165111 (2014). [CrossRef]
18. F. Benabid, J. C. Knight, G. Antonopoulos, and P. S. J. Russell, “Stimulated raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” Science 298(5592), 399–402 (2002). [CrossRef]
19. F. Couny, F. Benabid, P. J. Roberts, P. S. Light, and M. G. Raymer, “Generation and photonic guidance of multi-octave optical-frequency combs,” Science 318(5853), 1118–1121 (2007). [CrossRef]
20. 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]
21. S. T. Ha, Y. H. Fu, N. K. Emani, Z. Pan, R. M. Bakker, R. Paniagua-Dominguez, and A. I. Kuznetsov, “Directional lasing in resonant semiconductor nanoantenna arrays,” Nat. Nanotechnol. 13(11), 1042–1047 (2018). [CrossRef]
22. K. Koshelev, S. Kruk, E. Melik-Gaykazyan, J.-H. Choi, A. Bogdanov, H.-G. Park, and Y. Kivshar, “Subwavelength dielectric resonators for nonlinear nanophotonics,” Science 367(6475), 288–292 (2020). [CrossRef]
23. D. R. Abujetas, A. Barreda, F. Moreno, A. Litman, J.-M. Geffrin, and J. A. Sánchez-Gil, “High-Q transparency band in all-dielectric metasurfaces induced by a quasi bound state in the continuum,” Laser Photonics Rev. 15(1), 2000263 (2021). [CrossRef]
24. 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]
25. 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]
26. S. Chen, Z. Chen, J. Liu, J. Cheng, Y. Zhou, L. Xiao, and K. Chen, “Ultra-narrow band mid-infrared perfect absorber based on hybrid dielectric metasurface,” Nanomaterials 9(10), 1350 (2019). [CrossRef]
27. C. W. Hsu, B. Zhen, J. Lee, S.-L. Chua, S. G. Johnson, J. D. Joannopoulos, and M. Soljacic, “Observation of trapped light within the radiation continuum,” Nature 499(7457), 188–191 (2013). [CrossRef]
28. F. Monticone and A. Alu, “Embedded photonic eigenvalues in 3d nanostructures,” Phys. Rev. Lett. 112(21), 213903 (2014). [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. R. Parker, “Resonance effects in wake shedding from parallel plates: some experimental observations,” J. Sound Vibr. 4(1), 62–72 (1966). [CrossRef]
31. F. Dreisow, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tunnermann, and S. Longhi, “Adiabatic transfer of light via a continuum in optical waveguides,” Opt. Lett. 34(16), 2405–2407 (2009). [CrossRef]
32. M. Liu, D. A. Powell, R. Guo, I. V. Shadrivov, and Y. S. Kivshar, “Polarization-induced chirality in metamaterials via optomechanical interaction,” Adv. Opt. Mater. 5(16), 1600760 (2017). [CrossRef]
33. M. Liu and D.-Y. Choi, “Extreme huygens’ metasurfaces based on quasi-bound states in the continuum,” Nano Lett. 18(12), 8062–8069 (2018). [CrossRef]
34. 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]
35. J. Tian, Q. Li, P. A. Belov, R. K. Sinha, W. Qian, and M. Qiu, “High-Q all-dielectric metasurface: super and suppressed optical absorption,” ACS Photonics 7(6), 1436–1443 (2020). [CrossRef]
36. V. R. Tuz, V. V. Khardikov, A. S. Kupriianov, K. L. Domina, S. Xu, H. Wang, and H.-B. Sun, “High-quality trapped modes in all-dielectric metamaterials,” Opt. Express 26(3), 2905–2916 (2018). [CrossRef]
37. 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]
38. A. Forouzmand and H. Mosallaei, “All-dielectric c-shaped nanoantennas for light manipulation: Tailoring both magnetic and electric resonances to the desire,” Adv. Opt. Mater. 5(14), 1700147 (2017). [CrossRef]
39. S. Campione, S. Liu, L. I. Basilio, L. K. Warne, W. L. Langston, T. S. Luk, J. R. Wendt, J. L. Reno, G. A. Keeler, I. Brener, and M. B. Sinclair, “Broken symmetry dielectric resonators for high quality factor Fano metasurfaces,” ACS Photonics 3(12), 2362–2367 (2016). [CrossRef]
40. P. P. Vabishchevich, S. Liu, M. B. Sinclair, G. A. Keeler, G. M. Peake, and I. Brener, “Enhanced second-harmonic generation using broken symmetry iii-v semiconductor Fano metasurfaces,” ACS Photonics 5(5), 1685–1690 (2018). [CrossRef]
41. Z. Liu, Y. Xu, Y. Lin, J. Xiang, T. Feng, Q. Cao, J. Li, S. Lan, and J. Liu, “High-Q quasibound states in the continuum for nonlinear metasurfaces,” Phys. Rev. Lett. 123(25), 253901 (2019). [CrossRef]
42. A. B. Evlyukhin, S. I. Bozhevolnyi, A. Pors, M. G. Nielsen, I. P. Radko, M. Willatzen, and O. Albrektsen, “Detuned electrical dipoles for plasmonic sensing,” Nano Lett. 10(11), 4571–4577 (2010). [CrossRef]
43. F. Zhang, X. Huang, Q. Zhao, L. Chen, Y. Wang, Q. Li, X. He, C. Li, and K. Chen, “Fano resonance of an asymmetric dielectric wire pair,” Appl. Phys. Lett. 105(17), 172901 (2014). [CrossRef]
44. W. X. Lim and R. Singh, “Universal behaviour of high-Q Fano resonances in metamaterials: terahertz to near-infrared regime,” Nano Convergence 5(1), 5 (2018). [CrossRef]
45. D. R. Abujetas, N. Van Hoof, S. ter Huurne, J. G. Rivas, and J. A. Sanchez-Gil, “Spectral and temporal evidence of robust photonic bound states in the continuum on terahertz metasurfaces,” Optica 6(8), 996–1001 (2019). [CrossRef]
46. K. Koshelev, A. Bogdanov, and Y. Kivshar, “Meta-optics and bound states in the continuum,” Sci. Bull. 64(12), 836–842 (2019). [CrossRef]
47. 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]
48. V. V. Khardikov, E. O. Iarko, and S. L. Prosvirnin, “Trapping of light by metal arrays,” J. Opt. 12(4), 045102 (2010). [CrossRef]
49. R. Singh, I. A. I. Al-Naib, Y. Yang, D. R. Chowdhury, W. Cao, C. Rockstuhl, T. Ozaki, R. Morandotti, and W. Zhang, “Observing metamaterial induced transparency in individual Fano resonators with broken symmetry,” Appl. Phys. Lett. 99(20), 201107 (2011). [CrossRef]
50. M. Zhang, F. Zhang, M. Li, P. Li, J. Li, J. Li, and D. Yang, “Highly efficient amplitude modulation of terahertz Fano resonance based on si photoactive substrate by low power continuous wave,” Adv. Mater. Technol. 5(12), 2000626 (2020). [CrossRef]
51. C. Ma, Q. Lin, L. Wang, and K. Huang, “Highly tunable dual bound states in the continuum in bulk dirac semimetal metasurface,” Appl. Phys. Express 14(4), 042002 (2021). [CrossRef]
52. 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]
53. Z. F. Sadrieva, M. A. Belyakov, M. A. Balezin, P. V. Kapitanova, E. A. Nenasheva, A. F. Sadreev, and A. A. Bogdanov, “Experimental observation of a symmetry-protected bound state in the continuum in a chain of dielectric disks,” Phys. Rev. A 99(5), 053804 (2019). [CrossRef]
54. E. N. Bulgakov and D. N. Maksimov, “Avoided crossings and bound states in the continuum in low-contrast dielectric gratings,” Phys. Rev. A 98(5), 053840 (2018). [CrossRef]
55. S. Han, P. Pitchappa, W. Wang, Y. K. Srivastava, M. V. Rybin, and R. Singh, “Extended bound states in the continuum with symmetry-broken terahertz dielectric metasurfaces,” Adv. Opt. Mater. 9(7), 2002001 (2021). [CrossRef]
56. C. Kyaw, R. Yahiaoui, J. A. Burrow, V. Tran, K. Keelen, W. Sims, E. C. Red, W. S. Rockward, M. A. Thomas, A. Sarangan, I. Agha, and T. A. Searles, “Polarization-selective modulation of supercavity resonances originating from bound states in the continuum,” Commun. Phys. 3(1), 212 (2020). [CrossRef]
57. X. Zhao, C. Chen, K. Kaj, I. Hammock, Y. Huang, R. D. Averitt, and X. Zhang, “Terahertz investigation of bound states in the continuum of metallic metasurfaces,” Optica 7(11), 1548–1554 (2020). [CrossRef]
58. J. Li, J. Li, C. Zheng, Z. Yue, S. Wang, M. Li, H. Zhao, Y. Zhang, and J. Yao, “Free switch between bound states in the continuum (bic) and quasi-bic supported by graphene-metal terahertz metasurfaces,” Carbon 182, 506–515 (2021). [CrossRef]
59. J. Niu, Y. Zhai, Q. Han, J. Liu, and B. Yang, “Resonance-trapped bound states in the continuum in metallic thz metasurfaces,” Opt. Lett. 46(2), 162–165 (2021). [CrossRef]
60. R.-L. Chern, Y.-T. Chen, and H.-Y. Lin, “Anomalous optical absorption in metallic gratings with subwavelength slits,” Opt. Express 18(19), 19510–19521 (2010). [CrossRef]
61. Y.-T. Chen, R.-L. Chern, and H.-Y. Lin, “Multiple Fano resonances in metallic arrays of asymmetric dual stripes,” Appl. Opt. 49(15), 2819–2826 (2010). [CrossRef]
62. X. Luo, X. Li, T. Lang, X. Jing, and Z. Hong, “Excitation of high q toroidal dipole resonance in an all-dielectric metasurface,” Opt. Mater. Express 10(2), 358–368 (2020). [CrossRef]
63. B. Yuan, Y. Cheng, and X. Liu, “Conversion of sound radiation pattern via gradient acoustic metasurface with space-coiling structure,” Appl. Phys. Express 8(2), 027301 (2015). [CrossRef]
64. Q. W. Lin and H. Wong, “A low-profile and wideband lens antenna based on high-refractive-index metasurface,” IEEE Trans. Antennas Propag. 66(11), 5764–5772 (2018). [CrossRef]
65. S. R. Amanaganti, M. Ravnik, and J. Dontabhaktuni, “Collective photonic response of high refractive index dielectric metasurfaces,” Sci. Rep. 10(1), 15599 (2020). [CrossRef]
66. A. Harumi, K. Endo, and T. Suzuki, “Reflectionless metasurface with high refractive index in the terahertz waveband,” Opt. Express 29(10), 14513–14524 (2021). [CrossRef]
67. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60(8), 5751–5758 (1999). [CrossRef]
68. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001). [CrossRef]
69. T.-I. Weng and G. Y. Guo, “Band structure of honeycomb photonic crystal slabs,” J. Appl. Phys. 99(9), 093102 (2006). [CrossRef]
70. T. F. Khalkhali, B. Rezaei, and A. H. Ramezani, “Tuning of full band gap in anisotropic photonic crystal slabs using a liquid crystal,” Opt. Commun. 285(24), 5254–5258 (2012). [CrossRef]
71. T. Christopoulos, O. Tsilipakos, G. Sinatkas, and E. E. Kriezis, “On the calculation of the quality factor in contemporary photonic resonant structures,” Opt. Express 27(10), 14505–14522 (2019). [CrossRef]
72. D. M. Pozar, Microwave Engineering (Wiley, 2011), 4th ed.
73. R. Singh, I. A. I. Al-Naib, M. Koch, and W. Zhang, “Asymmetric planar terahertz metamaterials,” Opt. Express 18(12), 13044–13050 (2010). [CrossRef]
74. R. Singh, I. A. I. Al-Naib, M. Koch, and W. Zhang, “Sharp Fano resonances in thz metamaterials,” Opt. Express 19(7), 6312–6319 (2011). [CrossRef]
75. L. Cong, M. Manjappa, N. Xu, I. Al-Naib, W. Zhang, and R. Singh, “Fano resonances in terahertz metasurfaces: a figure of merit optimization,” Adv. Opt. Mater. 3(11), 1537–1543 (2015). [CrossRef]
76. M. Manjappa, S.-Y. Chiam, L. Cong, A. A. Bettiol, W. Zhang, and R. Singh, “Tailoring the slow light behavior in terahertz metasurfaces,” Appl. Phys. Lett. 106(18), 181101 (2015). [CrossRef]
77. Y. K. Srivastava, M. Manjappa, L. Cong, W. Cao, I. Al-Naib, W. Zhang, and R. Singh, “Ultrahigh-Q Fano resonances in terahertz metasurfaces:strong influence of metallic conductivity at extremely low asymmetry,” Adv. Opt. Mater. 4(3), 457–463 (2016). [CrossRef]
78. J. L. Pura, R. Kabonire, D. R. Abujetas, and J. A. Sánchez-Gil, “Tailoring polarization conversion in achiral all-dielectric metasurfaces by using quasi-bound states in the continuum,” Nanomaterials 12(13), 2252 (2022). [CrossRef]
