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Abstract
Optical metasurfaces with high quality factors (Q-factors) of bound states in the continuum (BICs) can substantially boost light–matter interaction for various applications in ultrathin and active metadevices. In this paper, we propose a metasurface supporting both at $\Gamma$ BICs and off $\Gamma$ BICs, and they can be modulated regularly. The sensitivities of the monolayer and bi-layer structures can reach 157.918 nm/RIU and 165.76 nm/RIU, respectively, at incident angles of 0.01 deg. For the bi-layer metasurface, four BICs at $\Gamma$ point are achieved, and they are influenced by the structural parameters dramatically. Additionally, the four at $\Gamma$ BICs coincide into two and then behave similarly to the monolayer case with the distance being large enough, which is associated with the BICs becoming uncoupled when the layers get far from each other. Furthermore, similar behaviors (four at $\Gamma$ BICs coincide into two) are observed by varying the nanopore size in one layer of the bi-layer metasurface. For an oblique incidence, two off $\Gamma$ BICs show up, and their center wavelengths have a dependent relationship with the size of the nanopores and the distance between layers. The results for monolayer and bi-layer metasurfaces are useful for potential applications based on BICs, such as sensors and filters.
© 2023 Optica Publishing Group
1. INTRODUCTION
In recent years, nano-photonics has been an active field in cutting-edge optical research and has been continuously studied at the subwavelength scale. In quantum physics, the spectrum of an open system usually contains bound states with discrete energy levels and resonant or leaky states in the continuous spectrum [1]. The former is completely localized, with no energy radiation, while the latter can couple with the continuous states and have some energy radiation [1,2]. Friedrich and Wigner specifically demonstrated the existence of bound states in the continuum (BICs) in multi-electron atomic models through two carefully designed resonance experiments [3]. Hsu verified the existence of BICs through a photonic crystal plate [4]. As a result, the discovery of BICs has been successfully extended to more wave physics fields such as electromagnetic waves, sound waves, and water waves, building the BICs system [4–6]. The literature shows two approaches to realize a BIC in the metasurface, where the first is the accidental decoupling from the radiation continuum via continuous tuning of system parameters, and the second relies on the presence of structural symmetry. The two scenarios lead to two distinctly different kinds of BICs: accidental and symmetry-protected [7,8]. When the resonance structures of multiple radiation channels couple without other loss, the channels will interfere with each other and completely offset the formation of a Fabry–Perot cavity, and thus it can also be called Fabry–Perot BICs [1]. Compared to traditional large-scale structures, subwavelength microstructures can cause Mie resonances to have strong field localization capabilities and typically have low radiation loss. Through the clever design of microstructures, the field can be localized at the subwavelength scale, greatly promoting practical applications [9,10]. Based on the potential applications of BICs super-surfaces, there is potential for enhancement in areas such as optical sensors [11], stealth [12–14], optical integration [15,16], nonlinear photonics [17–22], high-Q chiroptical resonances [23–25], and micro-nano optics [26–30].
In the direction of BICs research, most literature is based on monolayer structure studies, including some quasi-BICs discussion [20,31] and quasi-BICs-supported EIT studies [32]. Algorri et al. investigated the nature of both BICs and non-BICs resonances supported by metasurfaces by employing the Cartesian multipole decomposition technique [33]. There are also studies on the BICs properties of multilayer structures [34]. These papers have made a very in-depth exploration of BICs applications and obtained valuable results. Basically, they focus on the exploration of a single BIC, and the exploration of multiple BICs for multilayer structures is still insufficient.
In this paper, multiple BICs of monolayer and bi-layer metasurfaces with four square nanopores in one lattice in each layer are studied under $p$-polarized plane waves. Two symmetry-protected BICs at $\Gamma$ point with normal incidence and one off $\Gamma$ BIC at arbitrary oblique incidence are obtained for the monolayer metasurface with Q-factors. All BICs blueshift with a side length of all the square nanopores rising. Simultaneously, when we break the symmetry of the corresponding four-nanopore array, Q-factors of the metasurface get lower. Off $\Gamma$ BICs are insignificantly influenced by the side length variation of one of the nanopores in each lattice. Furthermore, the sensitivities of the monolayer and bi-layer structures can reach 157.918 nm/RIU and 165.76 nm/RIU, respectively, at incident angles of 0.01 deg. For the bi-layer metasurface, four BICs at $\Gamma$ point are achieved, and they are influenced by the structural parameters dramatically. Additionally, the four BICs at $\Gamma$ point coincide into two, and then they behave similarly to the monolayer case when the distance between layers is large enough, which is associated with the BICs becoming uncoupled when the layers are far from each other. Furthermore, similar behaviors are observed by varying the nanopore size in one layer of the bi-layer. In this paper, the results are helpful in line shape engineering. For oblique incidence, two off $\Gamma$ BICs show up, and their center wavelengths have a dependent relationship with the size of the nanopores and the distance between layers. The results for the monolayer and bi-layer metasurfaces are useful for potential applications based on BICs, such as sensors and filters.
2. RESULTS
A. Establishment of the Formal Theory
We start the study of the evolution of modes in the model with the assistance of temporal coupled-mode theory (CMT) [35–37]. $A$ is the resonance field, ${s_{m \pm}}$ is the incoming/outgoing plane waves, ${\tau _r}$ and ${\tau _{{\rm nr}}}$ are the radiative-decay lifetime and non-radiative-decay lifetime ${\tau _{{\rm nr}}}$, respectively, ${k_1}$ and ${k_2}$ are coupling coefficients when incoming plane waves excite the resonance, ${t_{{\rm slab}}}$ and ${r_{{\rm slab}}}$ are transmission and reflectivity coefficients, respectively, and coupling coefficients of the resonance decaying into the outgoing plane waves are denoted by ${d_1}$ and ${d_2}$:
$\gamma = \frac{1}{{{\tau _{{r}}}}} + \frac{1}{{{\tau _{{\rm{nr}}}}}}$, ${d_1}{k_1} = f \gamma$, ${f}$ is the complex amplitude of the resonant mode, $f = {t_{{\rm slab}}} + {r_{{\rm slab}}}$. The reflected amplitude $r$ can be expressed as follows:
The overall reflectance can be written as
B. Design of the Monolayer Metasurface and BICs Controlling
We utilize the monolayer metasurface composed of a ${\rm{S}}{{\rm{i}}_3}{{\rm{N}}_4}$ slab with refractive index ${{{n}}_1} = {2.02}$ penetrated with four square nanopores in a lattice periodically immersed in ${\rm{Si}}{{\rm{O}}_2}$ with refractive index ${{{n}}_2} = {1.46}$. As illustrated in Fig. 1, periodicity and thickness of the structure in the whole paper are kept as ${{\rm{P}}_x} = {{\rm{P}}_y} = {0.336}\;\unicode{x00B5}{\rm m}$ and ${\rm{H}} = {0.18}\;\unicode{x00B5}{\rm m}$, respectively. A $p$-polarized plane wave incidents in ${-}z$ direction with polarization in $x$ direction. Side lengths of the nanopores are denoted as ${{\rm{b}}_1}$, ${{\rm{b}}_2}$, ${{\rm{b}}_3}$, and ${{\rm{b}}_4}$.
The size dimension of the nanopores on the metasurface plays a key role in tailoring the reflection spectra line shape. We numerically investigate the reflectance of a monolayer metasurface with periodic four-nanopore arrays by using simulation and compare it with theory. First, we focused on the monolayer structure with symmetrical parameters. The reflectance spectra near the $\Gamma$ point of the system are shown in Figs. 2(a)–2(c), with side lengths of the nanopores at 0.068 µm, 0.076 µm, and 0.08 µm, respectively. Two symmetry-protected BICs are obtained, which obviously blueshift with the increasing side length. For example, BIC-I blueshifts by 0.002 µm, while BIC-II blueshifts by 0.0024 µm, accompanied by a widening of the distance between the two at $\Gamma$ BICs, with a side length of each nanopore enlarging from 0.076 µm to 0.08 µm. The BICs are attributed to the elimination of far-field radiation due to the symmetrical design. The zero line width at $\Gamma$ points indicates that there is no coupling to any outgoing wave, as it is forbidden. The supercavity mode composed of this structure can effectively suppress radiation leakage [38,39].
Fig. 1. Schematic of metasurface (a) from a top view and (b) unit cell of the four square nanopores.
These BICs exist robustly with the side lengths of the four nanopores in each lattice varying simultaneously because the symmetry of the system is not broken. When the metasurface is exposed to an oblique incident wave, an accidental BIC is also observed at an off $\Gamma$ point, as shown in Figs. 2(d)–2(f), which is associated with the tuning of structure parameters and is influenced insignificantly by symmetry. The off $\Gamma$ BIC is located at a wavelength of 0.7365 µm and occurs at an incident angle of 24 deg when the side length of the four square nanopores is 0.076 µm. It moves to a shorter wavelength of 0.7292 µm and lower incident angle of 23.2 deg with the nanopore size increasing to 0.08 µm.
We extract the resonance lifetime around the at $\Gamma$ BICs from the Fano properties shown in Fig. 3 to refine the results in Fig. 2. The resonance lifetime can be determined from the reflectance spectra by extracting the spectral line width of the observed Fano characteristics.
Fig. 2. Reflectance spectra of the monolayer metasurface with side lengths (a), (d) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.068}\;\unicode{x00B5}{\rm m}$; (b), (e) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.076}\;\unicode{x00B5}{\rm m}$; and (c), (f) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$ of the four squares in one lattice.
Fig. 3. Reflectance spectra around (a) BIC-I and (b) BIC-II of the monolayer metasurface with ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$ at incident angles of 0.1 deg, 0.3 deg, and 0.5 deg by simulation (solid line) and time domain theory (dotted). Q-factor variation of (c) BIC-I and (d) BIC-II at $\Gamma$ point.
The reflectance spectra of the metasurface with four identical nanopores in each lattice (${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$) at incident angles of 0.1 deg, 0.3 deg, and 0.5 deg are plotted in Fig. 3(a) by simulation (solid line) and theory (dotted). All the Fano resonance peaks are ultra-narrow around a tiny deviation from normal incidence, and the full width at half maximum (FWHM) narrows to none gradually with the incident angle decreasing to zero; the results from both methods agree well. The Q-factor is defined as the ratio of resonance frequency to spectra width. The parameters ${\lambda _0}$, $q$, and $\gamma$ for theory results at different incident angles in Figs. 3(a) and 3(c) are shown in Table 1. Additionally, it is a little difficult to distinguish a quasi-BIC from a BIC in Fig. 2 and Figs. 3(a) and 3(b). It is necessary to plot the corresponding Q-factor of BICs. Therefore, we draw the Q-factor of BIC-I and BIC-II in Figs. 3(c) and 3(d). The Q-factor is small with a tiny oblique incidence; with deviation from normal incidence, however, it increases dramatically to a value exceeding ${{1}}{{{0}}^6}$ with the deviation diminishing to zero, which proves the existence of BIC-I and BIC-II.
Fig. 4. Reflectance spectra of the monolayer metasurface with side lengths (a), (d) ${{\rm{b}}_1} = {0.076}\;\unicode{x00B5}{\rm m}$; and (b), (e) ${{\rm{b}}_1} = {0.084}\;\unicode{x00B5}{\rm m}$ of one nanopore of the four in one lattice, and ${{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$. (c), (f) Side lengths ${{\rm{b}}_1} = {0.066}\;\unicode{x00B5}{\rm m}$, ${{\rm{b}}_2} = {0.072}\;\unicode{x00B5}{\rm m}$, ${{\rm{b}}_3} = {0.080}\;\unicode{x00B5}{\rm m}$, and ${{\rm{b}}_4} = {0.088}\;\unicode{x00B5}{\rm m}$.
We note that such an in-plane inversion symmetry-protected BICs deserves further analysis because, intuitively, the Fano resonance peak width is associated with the symmetricity of the metasurface. It is necessary to clarify the BICs and their variation in the reflectance spectra with partially broken symmetricity of the four-nanopore structure. We investigated the reflectance spectra of the monolayer metasurface with side lengths in Figs. 4(a) and 4(d) of ${{\rm{b}}_1} = {0.076}\;\unicode{x00B5}{\rm m}$, and 4(b) and 4(e) ${{\rm{b}}_1} = {0.084}\;\unicode{x00B5}{\rm m}$, while the other three stay as ${{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$. Two BICs are obtained at 0 deg for the metasurface in Fig. 4(a), accompanied by the wavelength of the BICs blueshifting and the distance between the two resonance modes getting broader; the width of reflectance spectra would increase due to the number of leaky modes that couple with the BICs, leading to higher energy leakage and radiation losses. The off $\Gamma$ BIC in the faint reflection line is not obviously influenced, as shown in Figs. 4(d) and 4(e). This off $\Gamma$ BIC phenomenon is due to the interaction of different radiation channels of the square nanopore array structure, and the coupling between multipoles that binds the energy in the local field.
It is worth discussing the reflectance properties of the four-nanopore structure when the symmetry is totally broken, as shown in Fig. 4(c). It is observed that the symmetry-protected BICs disappear with the lowest Q-factor. This is due to the fact that TE modes (transverse Mie resonance) and TM modes (longitudinal Fabry–Perot resonance) are orthogonal and do not interact inside the structure [40]. Outside the nanostructure, the radiation fields of TE and TM modes have a similar partial interference phase length, resulting in a higher radiation pattern. By breaking the symmetry of the structure, the interference of the reverse oscillation cancels, forming a sub-radiation mode (quasi-BICs mode) in a square nanopore array monolayer [41]. The leaky mode is formed by partially breaking the bound states in the continuous domain generated by the collective magnetic dipole resonance excited in the sub-diffraction periodic system [18].
To visually analyze the BICs, near-field distributions of the normalized ${{\boldsymbol{E}}_{\boldsymbol{z}}}$ electric fields inside a single lattice of the symmetry metasurface are plotted in Figs. 5(a) and 5(b). Through the normalization of the electric field, compared with the structured electric field with no structured electric field, the results show that the electric field enhancement has strengthened to ${2.26} \times {{1}}{{{0}}^6}$ times from ${{\boldsymbol{E}}_{\boldsymbol{0}}}$. The charge in the nano-square hole array photonic crystal material will produce polarization under the action of the electric field component, and the magnetic pole will produce magnetization under the action of the magnetic field component. Dynamic polarization and magnetization may also interact with each other, which will affect the electromagnetic field. When light illuminates the metasurface normally, the reflectance spectral width of the structure approaches zero, and no radiation can be observed. Therefore, when calculating the electric field distribution, the incident angle is set slightly deviating from the angle where BICs occurs. The square nanopore array structure is a simple spatial arrangement symmetric about the $x$ and $y$ axes. When incident light hits the structure normally, electric dipoles and magnetic dipoles will be bound to the surface and oscillate, forming a surface state. Moreover, the symmetry between the surface state and incident light does not match, so the surface state cannot be coupled with the outgoing light wave of the structure. Therefore, the light will not radiate outward. According to Fig. 5(b) in the micro-nano structure, BICs is formed because two or more radiation modes of the same frequency are eliminated by far-field interference outside the structure, resulting in a hybrid mode of sub-radiation.
Fig. 5. Normalized ${{\boldsymbol{E}}_{\boldsymbol{z}}}$ electric field distributions of the symmetric square nanopore array structure in the (a) $x - y$ cross section at ${\rm{z}} = {{0}}$, $\lambda = {0.5581}\;\unicode{x00B5}{\rm m}$, and incident angle $\theta = {0.01}\;{\deg}$, and (b) $x - y$ cross section at ${{z}} = {{0}}$, $\lambda = {0.7292}\;\unicode{x00B5}{\rm m}$, and incident angle $\theta = {23.2}\;{\deg}$. (c) Center wavelength and (d) sensitivity of the Fano resonance peak around BIC-I of monolayer and bi-layer structure as a function of the refractive index of the surrounding medium. The size dimension of the four nanopores is ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, and ${{\rm{H}}_1} = {0.18}\;\unicode{x00B5}{\rm m}$.
Table 2. Comparison of the Properties of Dielectric Metasurface
To confirm the dependent relationship between BICs and the surrounding medium, we plotted the center wavelength of the Fano resonance at an incident angle slightly deviating from the BICs as a function of the refractive index of the surrounding medium. Since the peak width is zero, it is difficult to determine the center wavelength precisely. As shown in Figs. 5(c) and 5(d), the center wavelengths of the Fano resonance peak near BICs distinctly redshift as the refractive index of the surrounding medium varies by a tiny value (central wavelength shifts 0.0142 µm at 0.01 deg in monolayer and 0.0149 µm in bi-layer, with the refractive index of the surrounding medium increasing from 1.4 to 1.49), presenting an approximately linear relationship. When $\theta = {0.01}\;{\deg}$ in monolayer and bi-layer, all of the above considerations in Figs. 5(c) and 5(d) are under the condition of ${{{n}}_2} = {1.4}$. Therefore, the monolayer and bi-layer structure sensitivities at BIC-I are ${{\rm{S}}_{{\rm monolayer}}} = {157.918}\;{\rm{nm/RIU}}$ and ${{\rm{S}}_{\rm bi - layer}} = {165.76}\;{\rm{nm/RIU}}$, respectively. We compare it to the sensitivities of other papers, as shown in Table 2. Furthermore, the figure of merit (FOM) of the metasurface around BIC-I is about ${190262.7}\;{{\rm{RIU}}^{- 1}}$ in the monolayer metasrface and ${141675.2}\;{{\rm{RIU}}^{- 1}}$ in the bi-layer, according to the formula ${\rm FOM} = \frac{{\Delta \lambda}/ {\Delta n}}{{\rm FWHM}}$ of the refractive index sensor [37]. This proves that the metasurface considered here is significant in potential sensor applications.
Fig. 6. Schematic diagram of the bi-layer metasurface of identical four-nanopore arrays. (a) Top view of the metasurface and (b) unit cell of the bi-layer structure.
Fig. 7. Reflectance spectra of the bi-layer metasurface with the distance between two layers: (a) ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$, (b) ${\rm{D}} = {0.33}\;\unicode{x00B5}{\rm m}$, (c) ${\rm{D}} = {0.5}\;\unicode{x00B5}{\rm m}$, and (d) ${\rm{D}} = {1.0}\;\unicode{x00B5}{\rm m}$ around normal incidence, and (e) ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$ and (f) ${\rm{D}} = {1.0}\;\unicode{x00B5}{\rm m}$ for oblique incidence, with side lengths of the nanopore ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, and ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$.
C. Design of the Bi-layer Metasurface and BICs Controlling
To improve the modulation depth of BICs, we study the photonic property of the bi-layer metasurface composed of two identical nanopore arrays, as shown in Fig. 6. The periodicity and thickness are set as ${{\rm{P}}_x} = {{\rm{P}}_y} = {0.336}\;\unicode{x00B5}{\rm m}$ and ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$, respectively. The distance between layers is denoted as ${\rm{D}}$, and side lengths are ${{\rm{b}}_1}$, ${{\rm{b}}_2}$, ${{\rm{b}}_3}$, and ${{\rm{b}}_4}$ in both layers. We expect to obtain distributions of the reflectance spectra in the bi-layer metasurface different from those in the monolayer one, in which two BICs are obtained at the $\Gamma$ point. The Fabry–Perot resonance between the two layers plays a key role in the success of the BICs number transition at $\Gamma$ point [7]. When we set the two identical layers parallel to each other with distance ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$, expectedly, four BICs are presented in the considered wavelength band, marked as BIC-I, BIC-II, BIC-III, and BIC-IV at the $\Gamma$ point, as shown in Fig. 7(a). We now numerically confirm this by altering the distance ${\rm{D}}$ between the two layers. BIC-I and BIC-II move to the lower wavelength, and BIC-III and BIC-IV move to the longer wavelength, with ${\rm{D}}$ increasing. When the distance is larger than ${\rm{D}} = {1.0}\;\unicode{x00B5}{\rm m}$, BIC-I and BIC-III coincide, with the same thing happening on BIC-II and BIC-IV, as shown in Figs. 7(a)–7(d). In addition, BIC-II has an electromagnetically induced transparency (EIT) due to the interaction of the sharp BIC resonance with the wide-band bright mode on the metasurface [32], as shown in Fig. 7(b). It means we can control the number of BICs not only by the layers of the metasurface, but also by altering the distance between layers. We can introduce more freedom to modulate the number of BICs and their locations in the spectrum line simply by using two identical layers without introducing more complex patterns in each layer.
Fig. 8. Q-factor of the bi-layer nanopore structure around (a) BIC-I, (b) BIC-II, (c) BIC-III, and (d) BIC-IV in Fig. 7(a), respectively, with the bi-layer structure designed as ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$, and ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$.
Besides the at $\Gamma$ BICs, off $\Gamma$ BICs are investigated in the bi-layer metasurface to clarify their behaviors. Figures 7(e) and 7(f) show the reflectance spectra of the bi-layer metasurface with the distances between the two layers ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$ and 1.0 µm, respectively. In contrast to the single ${\rm{off}} {\text-} \Gamma$ BIC observed under oblique incidence in the monolayer metasurface, two off $\Gamma$ BICs are clearly presented in Fig. 7(e) with ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$. For example, the upper off $\Gamma$ BIC denoted as BIC-I occurs at $\lambda = {0.7205}\;\unicode{x00B5}{\rm m}$, $\theta = {20.8}\;{\deg}$, and the lower one happens at $\lambda = {0.698}\;\unicode{x00B5}{\rm m}$, $\theta = {20.3}\;{\deg}$. With the distance between the two layers broadened to ${\rm{D}} = {1.0}\;\unicode{x00B5}{\rm m}$, as shown in Fig. 7(f), BIC-I changes insignificantly, but BIC-II becomes less obvious, accompanied by a resonance line that becomes less discernible.
To investigate the properties of the bi-layer structure, as shown in Fig. 8, we explored the Q-factor of the bi-layer structure for ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$, and ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$ around at $\Gamma$ BIC-I, BIC-II, BIC-III, BIC-IV in Fig. 7. We compared the results for the single-layer BIC-I and BIC-II modes and found that the highest Q-factor exceeds ${{1}}{{{0}}^7}$, which is of potential application in the design of photonic devices.
Fig. 9. Reflectance spectra of the bi-layer metasurface with a nanopore side length of one layer: (a) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.06}\;\unicode{x00B5}{\rm m}$, (b) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.07}\;\unicode{x00B5}{\rm m}$, (c) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.078}\;\unicode{x00B5}{\rm m}$, and (d) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$ around normal incidence, and (e) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.06}\;\unicode{x00B5}{\rm m}$ and (f) ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$ at oblique incidence, with side lengths of nanopores in the other layer ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, ${\rm{D}} = {1.0}\;\unicode{x00B5}{\rm m}$, and ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$.
To explore the optical properties effected by the parameters, we explore the influence on BICs of nanopore size of one layer of the bi-layer metasurface. As shown in Figs. 9(a)–9(d), we observe that as we gradually increase the size of nanopores in one layer, BIC-I approaches BIC-III, and BIC-II moves to BIC-IV; they coincide into two at $\Gamma$ BICs with size of nanopores increasing. Then, the two at $\Gamma$ BICs behave similarly to that of the monolayer metasurface. In addition, EIT phenomena were observed at BIC-I, as shown in Fig. 9(c). The results show the number of at $\Gamma$ BICs can be modulated by the nanopore size of one layer of the bi-layer metasurface. Figures 9(e) and 9(f) show the reflectance spectra under oblique incidence; two parallel lines containing off $\Gamma$ BICs would provide more ways to improve the BICs controlling freedom.
Fig. 10. Reflectance properties with ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, ${\rm{D}} = {0.25}\;\unicode{x00B5}{\rm m}$, ${{\rm{H}}_1} = {{\rm{H}}_2} = {0.18}\;\unicode{x00B5}{\rm m}$, and ${\rm{W}} = {{0}}\;\unicode{x00B5}{\rm m}$, 0.02 µm, and 0.4 µm, respectively. The inset shows a schematic diagram of the misalignments W between layers.
Fig. 11. Numerical reflectance spectra of the structure with (a) corner no passivation and (b) corner passivation for the monolayer metasurface, and (c) corner no passivation and (d) corner passivation of the bi-layer metasurface. The inset shows a structural schematic diagram. Structural parameters are the same which are ${{\rm{b}}_1} = {{\rm{b}}_2} = {{\rm{b}}_3} = {{\rm{b}}_4} = {0.08}\;\unicode{x00B5}{\rm m}$, ${\rm{R}} ={0.04}\;\unicode{x00B5}{\rm m}$, and ${{\rm{P}}_x} = {{\rm{P}}_y} = {0.336}\;\unicode{x00B5}{\rm m}$, respectively.
To estimate the influence of the fabrication imperfection of the metasurface, we explore the geometric fabrication discrepancies from the misalignments between layers and the corner passivation in Fig. 10 and Fig. 11, respectively. It is shown that the reflectance spectra and the BICs behave similarly to the perfect sample simulations, only with an insignificant shift. As a result, it proves the BICs of the metasurface are robust to fabrication imperfection.
3. CONCLUSION
In conclusion, we investigate the multiple BICs in monolayer and bi-layer all-dielectric metasurfaces under a $p$-polarized plane wave. Their wavelength locations and numbers can be modulated flexibly by regulating layers, distance between layers, and nanopore size in one layer of the metasurface. Specifically, we obtained multiple symmetry-protected BICs at $\Gamma$ point and multiple off $\Gamma$ BICs at arbitrary incidence. Additionally, they have a good tolerance for fabrication imperfection. In addition, the sensitivities of the monolayer and bi-layer structures can reach 157.918 nm/RIU and 165.76 nm/RIU, respectively, at incident angles of 0.01 deg. For the bi-layer metasurface, we achieve four at $\Gamma$ BICs; the BICs number and wavelength can be modulated flexibility by the structural parameters. For oblique incidence, two off $\Gamma$ BICs appeared, and their center wavelengths were dependent on the size of the nanopores and the distance between layers. The results for the monolayer and bi-layer metasurfaces have potential applications in BICs-based devices such as sensors and filters.
Funding
National Natural Science Foundation of China (11304094, 12172228); Natural Science Foundation of Hunan Province of China (2020JJ5153); Program of Foundation of Science and Technology Commission of Shanghai Municipality (22dz1204202); Natural Science Foundation of Shanghai (22ZR1444400); Horizontal Scientific Research Project (H-2021-304-049, H-2021-304-128).
Disclosures
The authors declare no conflicts of interest. The authors are responsible for the content and writing of this paper.
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.
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