Figure 1a shows the unit cell of the considered all-dielectric superlattice metasurface. It is composed of three identical nanobeams made of silicon (n = 3.47) with a length of 700 nm, width of 150 nm and height of 110 nm. The periods along the x- and y-direction are px = 700 nm, py = 750 nm, respectively. The displacement between adjacent nanobeams within each unit cell is defined as the local displacement (d1), while the displacement between nanobeams in adjacent unit cells is defined as the global displacement (d2). Here, we demonstrate the manipulation of quasi-BIC by introducing a displacement difference parameter (△d = d1-d2) to open a leaky channel of the BIC, which originates from the band folding of guided mode to the continuum. Figure 1b shows the transmission spectra with respect to different △d for the metasurface embedded in free space for TE-polarized normal incidences. For the △d = 0 (denoted as the red circle) case, the metasurface supports a BIC state with infinitely high Q-factors. The transmission spectrum manifests a smooth profile without a sharp dip due to the uncoupling between the external environment and the ideal BIC state. When △d ≠ 0 (denoted as green and blue circles), a narrow dip arises in the transmission spectra, which means the excitation of quasi-BIC state with a finite Q-factor. Noteworthily, another Fano resonance originated from magnetic dipole (MD) mode at shorter wavelengths does not vanish no matter how the spacing changes, and the bandwidth and the resonance peak position remain almost unchanged during the variation. The possible reason is that for TE-polarized incidences, the MD Fano resonance at shorter wavelengths is produced by the mutual coupling between adjacent nanobeams with each unit cell, which is only related to the local displacement, while the quasi-BIC resonance at the longer wavelength is caused not only by the coupling between nanobeams within each unit cell, but also the coupling between the three-nanobeam unit-cells, which are governed by both the local and global displacements, respectively. Similarly, the transmission spectra with variations of other structural parameters (px, py, wx, wy and h) are shown in Fig. S1 of Supplementary Information. As can be seen, those parameters only affect the peak position of double asymmetric Fano resonances, but cannot tune the linewidth to infinitesimal. Only by adjusting the relationship between local and global displacements (△d or px) can we control the linewidth or even make it vanish. Figure 1c manifests the Q-factor and the resonance wavelength of the quasi-BIC as functions of △d around a BIC state. One can see that the linewidth of the quasi-BIC dwindles rapidly as △d approaches zero, while the resonance peak shifts to the longer wavelength with the increasing of △d. The Q-factor of the BIC mode tends to infinity at △d = 0. We also calculated the band diagram and the Q-factor map of eigenmodes supported in the △d = 0 metasurface (See Fig. S2 in Supplementary Information for more details). As we can see, the BIC is originated from the resonance coupling between the forward and backward guided modes, reach their highest Q-factors at the Γ point where they intersect with each other. To consider the practical conditions, results of the superlattice metasurface on a glass (n = 1.5) substrate are shown in Fig. S3 of Supplementary Information. It can be seen that the Q-factor decreases with the existence of a substrate, since the substrate converts the BIC into a resonant state, leaking through the radiation channel in the substrate . However, it remains high Q-factors and can be easily adjusted to the desired values by the displacement difference △d.
We recognize the BIC modes by utilizing multipole expansions [42–48] and near-field electromagnetic patterns of the modes. Figure 2a shows the simulated transmittance and reflectance spectra of the superlattice metasurface with the displacement difference parameter △d = -70 nm, embedded in free space for TE-polarized normal incidences, where double asymmetric Fano resonances with high Q-factors can be observed at λ = 847 nm and 1148 nm. Mode 1 refers to the MD Fano resonance at 847 nm. Mode 2 refers to the quasi-BIC mediated by the parameter △d at 1148 nm. The spectral curves in Fig. 2a correspond to the specific case indicated by the dashed line of Fig. 1b. Figures 2b and 2c depict the extinction cross-section spectra by multipole expansions of the MD Fano resonance (resonance 1) and quasi-BIC resonance (resonance 2), respectively. We found that MD Fano resonance is indeed dominated by the MD mode, with smaller contributions from the electric quadrupole (EQ) mode. The quasi-BIC resonance is completely different, which is dominated by the toroidal (TD) mode, with smaller contributions from the magnetic quadrupole (MQ) mode, representing the radiation channel coupled to the external waves in our system. The locations of the strong scattering intensity of the multipoles are consistent with the spectra in Fig. 2a. By analyzing the near-field electromagnetic patterns of the modes, shown in Figs. 2d and 2e, we further confirm that the MD mode with antisymmetric distributions of polarization currents, accompanied by the coexistence of EQ mode with different strengths at λ = 847 nm. Furthermore, the quasi-BIC resonance corresponds to TD mode with the dominant direction perpendicular to the incident light, accompanied by MQ mode with different strengths at λ = 1148 nm.
At the condition of △d ≠ 0, another symmetry-protected BIC is accompanied with our previous mentioned displacement-mediated quasi-BIC in the superlattice metasurface, which can be transferred to finite-linewdith quasi-BICs by varying the incident angle. The transmission spectra with respect to the angle and wavelength of TE-polarized incidence is depicted in Fig. 3a. The curve indicated by the green circle is the displacement-mediated quasi-BIC and the curve indicated by the blue circle is the symmetry-protected quasi-BIC mediated by the incident angle. Several appealing features can be observed from Fig. 3a. First, the linewidth and position of the green circle are determined by the parameter △d, which is also the largest linewidth of the quasi-BIC in the whole transmission spectra. If the parameter △d = 0, the quasi-BIC will not emerge at any angles, which is completely confined. Second, for quasi-BICs (the blue circle) mediated by the incident angle, the line width of the quasi-BIC resonance absolutely vanishes because the system is completely symmetrical at normal incidences and does not couple with the external environment. We note that the maximum linewidth of the quasi-BIC produced by the incident angle is also determined by the parameter △d. If △d = 0, the quasi-BIC mediated by the incident angle will not appear in the transmission spectrum. With the increase of △d, this quasi-BIC (the blue circle) appears in the case of oblique incidences, and the maximum linewidth of this quasi-BIC in the transmission spectrum increases. In other words, the quasi-BIC mediated by incident angles has the same resonance wavelength of the quasi-BIC caused by the displacement, with large angular dispersion. Moreover, the maximum linewidths of two types of the quasi-BICs appear at different positions in the transmission spectra, but their maximum linewidths are nearly identical and determined by the parameter △d. The band diagram of the TE mode is displayed in Fig. 3b, revealing large angular dispersions of the two types of quasi-BICs modes. In this regard, the band structure for the quasi-BIC case as shown in Fig. 3b reveals modes in the radiative region with high dispersion similar to that for the true BIC with a zero displacement in Fig. S2. Both are highly dispersive, but they have different diffraction lines since the lattice constant differs. Also, this is consistent with that if △d = 0, the quasi-BIC will not emerge at any angles. Figures 3c and 3d show their electric near-field patterns at the incident angle of 2°. Although the positions of the two quasi-BICs are very close to each other, the electromagnetic modes of the two quasi-BICs are different, which can be accurately analyzed by multipole expansions. Therefore, the extinction cross-section spectra by multipole expansions of the two quasi-BICs at the angle of 2° are presented in Fig. S4 of Supplementary Information, revealing that the absorbing phenomenon can be largely attributed to the coexistence of dominant multipoles. The displacement-mediated quasi-BIC is dominated by the TD mode, coupled with concomitance of other multipoles including MD and MQ modes with distinct strengths, while the angle-mediated quasi-BIC is produced by the concomitance of predominant multipoles including MD and EQ modes with unequal strengths. Figures 3e-3 g show the transmission spectra at angles of 2°, 15° and 30°, respectively, demonstrating that the two quasi-BICs mediated by the incident angle shift in the opposite directions with the increase of incident angles, which are promising to develop multiple angular dispersion applications.
Apart from the TE-polarized incidence, now we show that the displacement-mediated quasi-BIC is also applicable for the TM-polarized incidence (Fig. 4). Figure 4a shows the transmission and reflection spectra with parameter △d = -70 nm, revealing that one of the double asymmetric Fano resonances can be completely confined and converted into BICs with infinite Q-factors (Fig. 4b). Resonance 3 refers to the quasi-BIC mediated by the parameter △d at 785 nm. Resonance 4 refers to another TD Fano resonance at 955 nm. The spectral curves in Fig. 4a correspond to the specified case indicated by the dashed line of Fig. 4b. The quasi-BIC near the wavelength of 785 nm can be obtained when the local displacement is tuned to be not equal to the global displacement(△d ≠ 0), that is, the simple lattice is transformed to a superlattice. When △d ≠ 0, a narrow dip arises in the transmission spectra, which means the excitation of quasi-BIC state, while the resonance position remains almost invariant. On the other hand, the spectra width of the TD Fano resonance near the wavelength of 955 nm is nearly steady with the change of △d. While the resonance position locates at the shortest wavelength at △d = 0, and shifts to longer wavelengths with the increasing of |△d|. More interestingly, we calculated extinction cross-section spectra by multipolar expansions for TM-polarized normal incidences as shown in Figs. 4c and 4d. It reveals that the dominant modes of the shorter-wavelength resonance modes and long-wavelength resonance modes are identical for TM and TE polarizations, but the intensity distribution is slightly different, with different directions of the mode due to the different polarizations. Figure 4c shows that the quasi-BIC near 785 nm mediated by the parameter △d is dominated by the MD mode, coupled with smaller contributions from the EQ mode, which is similar to the modes of MD Fano resonance caused by the TE polarization in Fig. 2b. Then through the analysis of the electromagnetic field distribution in Fig. 4e, it is found that the mode is along the y-direction for the TM-polarized incidence, while the mode in Fig. 2d is along the x-direction for the TE polarization. Different polarizations lead to different electromagnetic directions with identical modes. In addition, for the TM polarization (Fig. 4d), the Fano resonance near 955 nm is dominated by the TD mode, with smaller contributions from the MQ mode. Similarly, it can be seen from Figs. 4f and 2e, the electromagnetic modes of the two are almost identical, but the directions are perpendicular to each other in the xy plane. The variation of △d can change the resonance position of the TD mode, but cannot change the resonance position of the MD mode. To summarize, for TM-polarized incidences, the resonance linewidth of the quasi-BIC dominated by the MD mode changes with the parameter △d, but the resonance position is basically unchanged, while the TD Fano resonance does not change the linewidth, but changes the position with the parameter △d. The quasi-BIC dominated by the MD mode under TM-polarized incidences can not only be controlled by △d, but also by other parameters (px, py, wx, wy, h, θ0), as shown in Fig. S5 of Supplementary Information, which demonstrates large tunability of the TM-polarized BIC modes.