Bound states in the continuum enabling ultra-narrowband perfect absorption

The designed Si metasurface is illustrated in figure 1(a), which consists of a periodic square array of hollow nanodisks, placed on a SiO₂/Au substrate. To analyze our system, we calculate the eigenmode spectra at the Γ point by means of a three-dimensional finite-element-method (FEM) eigenfrequency solver with COMSOL Multiphysics. In the simulation, the refractive indices of Si and SiO₂ are set as 3.5 and 1.45, respectively; for the plasmonic metal involved, we use a Lorentz–Drude formula to model the permittivity dispersion of Au [27]. We calculate the effective eigenmodes in a unit cell of the metasurface with periodic boundary conditions in the two lattice-vector directions and perfectly matched layers at the boundaries in the out-of-plane direction. Figure 1(b) shows the calculated eigenmode spectra as a function of the SiO₂ spacer thickness s. It can be observed that as s increases, there exists an eigenmode with a minimum loss, manifesting the formation of a BIC. Notably, in a lossy case, the total Q factor (Q_t) is defined as $Q_{\text{t}}^{ - 1} = Q_{\text{r}}^{ - 1} + Q_{{\text{nr}}}^{ - 1}$, with Q_r and Q_nr being the radiative and non-radiative Q factors, respectively. While the Q_r at the BIC point is inclined to infinity, the Q_nr is finite and predominantly limited by the Ohmic loss from the metal. Thus, the Q factor for the BIC does not diverge in our system. Due to the fact that the BIC is accidentally decoupled from the radiation continuum by continuously adjusting structural parameter, it can be called the accidental BIC. More specifically, this type of bound state is referred to as the Fabry–Pérot BIC, the formation of which is based on two identical resonances coupled through a single radiation channel. To confirm our analysis, we give in figure 1(d) the electromagnetic-field distributions of the BIC, revealing its nature of the in-plane MD. Therefore, the BIC can be thought of as the result of the coupling between the MD mode excited in the Si nanoparticle and its mirror image induced by metal substrate.

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In the absence of Ohmic losses, this coupling can be characterized by the following Hamiltonian [8, 10, 28, 29]

$$\begin{equation} {\hat H = \left[ {\begin{array}{*{20}{c}} {{\omega _0}}&\kappa \\ \kappa &{{\omega _0}} \end{array}} \right] - i\frac{{{\gamma _r}}}{2}\left[ {\begin{array}{*{20}{c}} 1&{ - {e^{i\psi }}} \\ { - {e^{i\psi }}}&1 \end{array}} \right]} \end{equation} \tag{ 1 }$$

where ${\omega _0}$ and ${\gamma _r}$ are the resonant frequency and the radiative decay rate of individual mode, respectively; $\psi = kd$ is the phase shift between the two modes, where k is the propagation constant and d the distance between the two modes; is the coupling factor. For the in-phase 'dipole-image' pairs, the eigenvalue of $\hat H$ is $\omega = {\omega _0} + \kappa + i{\gamma _r}\left[ {{e^{i\psi }} - 1} \right]$. When $\psi$ is equal to an integer multiple of 2π, the eigenvalue becomes a real number and the corresponding eigenmode turns into a true BIC. Because $\psi$ is directly related to d, we open the leaky channel of the BIC by usually tuning the distance between the two modes. Besides, the radiation of this BIC can also be controlled by tuning other parameters such as the dimensions of the nanoparticle and the period of structure. This is because $k = n{\omega _0}/c$, where n is the effective refractive index between the two modes and c is the light speed in vacuum. The frequency of individual mode, ${\omega _0}$, is associated with the above-mentioned structural parameters and is thus another factor affecting the phase shift. It is worth mentioning that the optical behavior of our metasurface is very similar to that of an all-dielectric nanostructure consisting of two metasurfaces. Using the all-dielectric system, we comprehensively demonstrate the influence of the above-mentioned parameters on the BIC (see appendix).

Another important category of BICs is the symmetry-protected BIC. For periodic structures, if they are with the time-reversal and the in-plane inversion symmetry and the mode frequency is below the diffraction limit, symmetry-protected BICs can exist at high-symmetry points of the first Brillouin zone, for example at the Γ point, owing to the symmetry mismatch between their mode profiles and external propagating modes [10, 11, 30, 31]. Interestingly, breaking the structural symmetry provides an efficient way to trigger the transition of BICs to the quasi-BIC under the normal incidence of light. For our metasurface, a parameter δ is used to determine the amount of the structural symmetry-breaking, which is equal to the offset of the air hole from the center of the concentric system, as illustrated in the inset of figure 1(c). Figure 1(c) displays the eigenmode spectra as a function of δ, where the thickness of the spacer is s = 98 nm. It is observed that a BIC appears at δ = 0 and transforms into the quasi-BIC whose loss grows with increase of the offset value. Figure 1(e) shows the electromagnetic-field distributions for the BIC, which manifests that this type of BIC is characterized by an out-of-plane MD mode.

To demonstrate the perfect absorption in our BIC metasurface, we optimize these two main geometrical parameters (s and δ) using the temporal CMT [32, 33]. In this theory, the absorbance at a frequency $\omega$ is expressed as:

$$\begin{equation} {A = \frac{{4{\gamma _{\text{r}}}{\gamma _{{\text{nr}}}}}}{{{{\left( {\omega - {\omega _0}} \right)}^2} + \gamma _t^2}}} \end{equation} \tag{ 2 }$$

where ${\gamma _{\text{r}}}$ and ${\gamma _{{\text{nr}}}}$ account for the radiative and the non-radiative damping rate, respectively; ${\gamma _t} = {\gamma _{\text{r}}} + {\gamma _{{\text{nr}}}}$ is the total damping rate. According to the equation, at the condition of critical coupling where ${\gamma _{\text{r}}} = {\gamma _{{\text{nr}}}}$, light with frequency ${\omega _0}$ can be totally absorbed. As shown in figure 2(a), we calculate the eigenfrequencies ${\omega _0} + i{\gamma _t}$ (indicated by solid lines) for the Fabry–Pérot BIC modes for different value of s. In order to determine ${\gamma _{\text{r}}}$, we consider a lossless case where the imaginary part of Au permittivity is assumed to be zero, and obtain corresponding eigenfrequencies ${\omega _0} + i{\gamma _{\text{r}}}$ (a short-dashed line). Then, the non-radiative loss is calculated by ${\gamma _{{\text{nr}}}} = { }{\gamma _t} - {\gamma _{\text{r}}}$. As can be seen, the critical coupling occurs at s = 83 nm and 113 nm, respectively, at which light coupled to the eigenmode is no longer scattered into space, but totally dissipated in the metal through Ohmic loss. This is because when placing the dielectric nanostructure onto an optically thick metal substrate, the transmission can be blocked and the reflection is canceled through the complete destructive interference with light coupled to quasi-BIC. Figure 2(b) shows the maximum peak absorption and total Q factor for the quasi-BIC modes as a function of s, from which one can see that the absorbance is close to 100% at the condition of critical coupling. Having determined the radiative and non-radiative losses under the critical-coupling condition (${\tilde \omega _0} = 236 + 0.22i\;{\text{THz}}$ for s = 83 nm and for s = 113 nm), the absorption spectrum can be obtained using the CMT, as shown by the black solid lines in figures 2(c) and (d). The results agree well with those calculated by FEM (green dots).

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Next, we turn our attention to the perfect absorption in asymmetric metasurfaces. We fix the spacer thickness as s = 113 nm and calculate the maximum absorbance varying with the asymmetric parameter δ. As shown in figure 3(a), because the Fabry–Pérot BIC (BIC #1) is much less sensitive to symmetry-breaking perturbations, the associated quasi-BIC resonance always remains absorbance close to 100% with the increase of δ. However, for the symmetry-protected BIC (BIC #2), there exists an optimized value of δ able to maximize absorption of the quasi-BIC mode, at which (δ = 13 nm) the critical coupling occurs. As shown in the inset of figure 3(a), we calculate δ-dependence damping rates for the eigenmode related to BIC #2, from which one can see that ${\gamma _{\text{r}}}$ equals to ${\gamma _{{\text{nr}}}}$ when δ = 13 nm. The resultant absorption spectrum is shown in figure 3(b), which exhibits two clear resonances with near-unity absorbance. Such dual-band perfect absorption is particularly favorable to a number of promising applications like multiplexing detector arrays, selective optical filters and biochemical sensing.

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From the absorption spectrum, some important features can be concluded. First, the two resonant modes separated by about 146 nm do not show the high degree of symmetry in the characteristics (that is, same absorption amplitudes and resonance linewidths). And evidently, the resonant mode associated with BIC #2 exhibits a narrower spectral linewidth due to higher Q factors of the symmetry-protected quasi-BIC (see figures 1(b) and (c)). Another feature is that the two modes exhibits different sensitivity to the polarization of incident light. This is shown in the inset of figure 3(b), where the absorption is plotted for a same nanostructure under normal incidence with polarization along y direction. It is obvious that the symmetry-protected BIC shows strong polarization dependence. In contrast, both the Q factor and resonant frequency for the Fabry–Pérot BIC are independent on the polarization of light, which is desirable for the applications using non-polarized light such as imaging and photodetection.

Up to now, we have studied how to manipulate the radiative losses of BIC modes for achieving the perfect absorption effect. Herein, we investigate the influence of the non-radiative decay on the absorption features of quasi-BIC resonances. In our systems, the non-radiative losses can be controlled by uniformly scaling the imaginary part ε'' of the permittivity of the Au substrate. As shown in figure 4(a), we take the Fabry–Pérot BIC mode in symmetric metasurfaces as an example, and calculate the damping rates of system. It can be seen that the non-radiative loss decreases as ε'' is reduced and the critical couplings are obtained at s = 111 nm, 110 nm, 108 nm and 106 nm, respectively. We also calculate the Q factor of system as a function of the thickness of SiO₂, as shown in figure 4(b), which is substantially increased when the non-radiative loss decreases to 0.2ε''. This case can lead to narrower linewidths for the perfect absorption spectra. To confirm this, we calculate the normally-incident absorption spectra of the metasurface at the conditions of critical coupling, as shown in figure 4(c). One can see that when the imaginary part of the permittivity of bulk Au is reduced to 0.1ε'', the absorption bandwidth less than 1 nm can be obtained for the metasurface. Thus, this narrowing mechanism driven by the non-radiative loss offers an efficient way to tailor the bandwidth of the metasurface absorbers. In practice, plasmonic materials, such as Ag and Al, can be employed as the material constituent of substrate for improving the Q factor of the perfect absorption peak owing to their fewer interband losses than Au at near-infrared and visible frequencies.

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