Observation of tunable accidental bound state in the continuum in silicon nanodisk array

2.1 Spectral and near-field characteristics

Figure 1(A) illustrates the periodic array of silicon nanodisks with diameter d, height h = 200 nm, and square periodicity Λ. The nanodisk array embedded in homogeneous dielectric environment with refractive index n ₀ = 1.45 is illuminated by plane wave with incidence angle θ and electric field polarized in the x–z plane and of unitary amplitude (|E ₀| = 1). Note that in this work, θ means the incidence angle in free space.

We adopted a home-built package based on rigorous coupled-wave analysis developed following [40], [41], [42] to simulate the zeroth-order transmittance spectra and the near-field distributions of the periodic silicon nanodisks. In all the simulations, the wavelength-dependent refractive indices of silicon were taken from Palik [43].

Figure 2(A) depicts the simulated angular-resolved transmittance spectra of the silicon nanodisk array with Λ = 555 nm and d = 535 nm. Strictly speaking, the structure under study is in the photonic crystal phase rather than in the metasurface phase based on the physics of Mie and Bragg resonances [44] (Figures S1 in the Supplementary). Results show that for θ = 10°, there exist five transmittance dips within the spectral range between 1.0 μm and 1.45 μm. When the first two dips or the last two dips are spectrally overlapped around θ = 20° or θ = 26°, respectively, nearly unitary transmittance and thus nearly zero reflectance can be obtained. This suggests multiple resonant lattice Kerker effects, which can be treated as the extension of our previous work [45]. This phenomenon is out of the scope of this work and will be reported elsewhere.

Figure 2:

Spectral, near-field and far-field characteristics of the A-BIC. (A) Simulated angular-resolved transmittance spectra of the silicon nanodisk array. (B) Zoom of the red box in (A). Red circles in (A) and (B) indicate the occurrence of the A-BIC at the position of (λ ₀ = 1.164 μm, θ ₀ = 25°). (C) Extracted quality factors (open circles) of the resonance shown in (B) as functions of θ. Solid curves denote theoretical fitting with inverse square functions. (D) and (E) Near-field electric filed distributions |E/E ₀|² (color for intensity and arrows for directions) for the A-BIC at (λ ₀, θ ₀), in (D) the x–z plane with y = 0 for the central panel, and (E) in the y–z plane with x = 0 for the bottom panel. Silicon nanodisks are outlined by white boxes. (F) Multipolar contents of the electromagnetic fields at the A-BIC. “ED,” “MD,” “EQ,” “MQ,” and “Sum” stand for electric dipole, magnetic dipole, electric quadrupole, magnetic quadrupole, and sum of all these multipoles, respectively.

Interestingly, we find that as θ increases, the third dip first gradually narrows down, disappears around θ ₀ = 25°, as indicated by the open circle, and emerges again with gradually increasing linewidth. This is better visualized by the zoomed view in Figure 2(B). We further extracted the Q factors and plot them as functions of the incidence angle in Figure 2(C). Results show that as θ approaches 25°, the Q factor increases significantly and tends to infinity. Indeed, the dependence of these Q factors on θ is inverse quadratic, as shown by the good agreement between the simulated values and the fitting curves, which follow

Therefore, the spectral behaviors, as well as the inverse quadratic relationships, suggest the occurrence of an A-BIC at θ ₀ = 25°, and the disappearance of the transmittance dip with infinite Q factor suggests that, the A-BIC is an “ideal” BIC in theory [46].

In order to understand the origin of this A-BIC with clear physical intuition, we turn to the near-field optical pictures and the multipolar decomposition. We emphasize here that the physical understanding of A-BIC based on the multipolar decomposition of electromagnetic fields is acceptable not only in metasurfaces but also in photonic crystals [7]. Figure 2(D) shows that, for the A-BIC at (λ ₀ = 1.164 μm, θ ₀ = 25°), the electric field in the x–z plane is mainly confined to the gaps between the silicon nanodisks, and its circulation within a unit cell forms an in-plane magnetic dipole (MD). On the other hand, Figure 2(E) shows that the electric field in the y–z plane is mainly confined to the central outside of the silicon nanodisks, forming an evident out-of-plane electric dipole (ED). The multipolar decomposition in Figure 2(F) verifies that dominant contributing contents are the ED and the MD, accompanied with relatively weak electric quadrupole (EQ) and magnetic quadrupole (MQ), which slightly complex the near-field distributions in Figure 2(D) and (E). Therefore, both the near-field distributions and the multipolar decomposition have revealed that the formation of the A-BIC can be understood by the approximately destructive interference of the dominating out-of-plane electric dipole and in-plane magnetic dipole, and weak electric and magnetic quadrupoles, which is similar to the literature [7].

As the incidence angle varies, the relative strengths of the multipolar contents vary (Figure S2 in the Supplementary). Nevertheless, the dominant contributing multipoles remain the out-of-plane ED and the in-plane MD, as illustrated by Figure 1(B). Therefore, the spectral and angular position of the A-BIC, (λ ₀, θ ₀), can be approximately determined by solving the following conditions expressed as [39]

which should be satisfied simultaneously. Here, G _bij are the matrix elements of the lattice depolarization Green dyadic G ↔ b , and α y ( m ) and α z ( e ) are the polarizabilities of the MD along the y axis and the ED along the z axis.

2.2 Experimental validations

To verify our numerical findings, we fabricated periodic silicon nanodisks on a quartz substrate using the state-of-art nanofabrication techniques, which involve the electron-beam lithography and deposition, liftoff, dry etching, and wet etching processes. The as-fabricated samples, of which a typical scanning electron image is shown in Figure 3(A), were then covered with index-matching oil in order to guarantee homogeneous dielectric environment surrounding the silicon nanodisks. The detailed descriptions on the fabrication and the optical characterization can be found in our previous work [47].

Figure 3:

Experimental demonstration of the A-BIC. (A) SEM image and (B) measured transmittance spectra of as-fabricated silicon nanodisk array with Λ = 555 nm and d = 535 nm. (C) Measured Q factors of quasi-A-BICs, i.e., the resonances corresponding to the nearly vertical transmittance dips around the red circle, as functions of θ. Solid curve denotes theoretical fitting with inverse square functions.

Figure 3(B) plots the measured angular-resolved transmittance spectra for θ varying from 0° to 40°. Comparing Figure 2(A) and (B), we find that in general, the measured transmittance spectra agree well with the simulation results. Remarkably, as θ increases from 0° to 40°, the nearly vertical transmittance dip first narrows down, disappears around θ ₀ = 20°, which is slightly smaller than the theoretical position (θ ₀ = 25° in Figure 2(A)), and emerges again with increasing linewidth. This suggests the occurrence of the A-BIC, as indicated by the red circle.

We further extracted the quality factors of the quasi-A-BIC around θ ₀ = 20° from the measured transmittance spectra in Figure 3(B). Figure 3(C) shows that the quality factor tends to increase significantly as θ approaches 20°. When θ becomes too close to 20°, the transmittance dip has so small modulation depth that it is difficult to extract the quality factor from the transmittance spectra. The dependence of the measured quality factor on the incidence angle also follows the inverse quadratic relationships that can be expressed in Equation (1). These behaviors are similar to the simulation results shown in Figure 2(C) except that the measured quality factors are smaller than the simulated results. The degraded quality factors in experiments may attribute to the finite size of samples, material absorption, surface scattering [46], and fabrication imperfections, which include variations in the nanodisks’ sizes and shapes as shown by Figure 3(A), and side-wall tilting, as well as the use of incoherent source in measurements, which greatly reduces the measurable Q factors, as demonstrated by Bin-Alam et al. [48]. Therefore, we have experimentally validated the occurrence of the A-BIC in the periodic silicon nanodisks and shown that it is not an “ideal” BIC in practice.

2.3 Tuning A-BIC via nanodisk size or lattice period

By decreasing the nanodisk diameter to d = 510 nm while keeping the lattice period to be Λ = 555 nm, we plot the simulated and measured angular-resolved transmittance spectra in Figure 4(A) and (B), respectively. Results show that for the third resonance, the A-BIC occurs at θ ₀ = 22° and λ ₀ = 1.138 μm in simulations, or θ ₀ = 17° and λ ₀ = 1.134 μm in experiments. Comparing with the results for d = 535 nm in Figure 2(A) and (B), we find that the spectral and angular position (λ ₀, θ ₀) for the A-BIC is shifted to the left bottom for both the simulation and the experimental results.

Figure 4:

Tuning the A-BIC via the nanodisk diameter. (A) Simulated and (B) measured angular-resolved transmittance spectra of silicon nanodisk array with Λ = 555 nm and d = 510 nm. (C) and (D) Comparison of simulated (circles) and measured (balls) results on the (C) spectral and (D) angular positions for the A-BICs as functions of the nanodisk diameter d.

We summarize the extracted spectral and angular positions for the A-BICs from the simulated and measured angular-resolved transmittance spectra for different nanodisk diameters (Figures S3 and S4 in the Supplementary) and plot the results in Figure 4(C) and (D). Results show that the spectral and angular position (λ ₀, θ ₀) for the A-BIC keeps increasing with the nanodisk diameter, consistent with the results in [39]. This behavior is experimentally demonstrated by the good agreement between the simulated and measured positions of (λ ₀, θ ₀). Strikingly, when the diameter decreases to d = 430 nm, θ ₀ can even reduce to 0°. This suggests an interesting transition of the A-BIC from the off-Γ-BIC to the at-Γ-BIC, which is a new phenomenon that has not been reported yet.

Similarly, the spectral and angular position (λ ₀, θ ₀) for the A-BIC also decreases with the nanodisk’s height (Figure S5 in the Supplementary). This behavior is also consistent with the results in [39].

Besides the nanodisk diameter and height, we find that (λ ₀, θ ₀) for the A-BIC can also be tuned by varying the lattice period, which was not shown in [39]. Figure 5(A) shows that if the lattice period increases to Λ = 575 nm while keeping the nanodisk diameter to be d = 535 nm, the angular position for the A-BIC decreases to θ ₀ = 22°, whereas the spectral position increases to λ ₀ = 1.168 μm. The simulated angular-resolved transmittance spectra are also validated by the experimental results, as shown by Figure 5(B), where good agreement can be found except that the measured position for the A-BIC (λ ₀, θ ₀) is also slightly smaller than the simulated data.

Figure 5:

Tuning the A-BIC via the lattice period. (A) Simulated and (B) measured angular-resolved transmittance spectra of silicon nanodisk array with Λ = 575 nm and d = 535 nm. (C) and (D) Comparison of simulated (circles) and measured (balls) results on the (C) spectral and (D) angular positions for the A-BICs as functions of the lattice period Λ.

By further increasing Λ, we find that λ ₀ keeps increasing, whereas θ ₀ keeps reducing, as shown by Figures 5(C) and (D) and S6 and S7 in the Supplementary. Similarly to Figure 4(D), we also find that θ ₀ can even reduce to 0° when Λ = 685 nm. In other words, the transition of the A-BIC from the off-Γ-BIC to the at-Γ-BIC can also be observed by increasing the lattice period.

Although the above behaviors of the spectral and angular position for the A-BIC as the nanodisk’s size or the lattice period varies can be approximately predicted by the theoretical expressions of Equations (2) and (3) that are developed in [39], here we turn to qualitative explanations with more straightforward physical pictures. For smaller nanodisk’s diameter or height or larger lattice period, the delocalized nature of the multipoles becomes more important. Therefore, if the nanodisk’s diameter or height decreases, the resonance wavelength is blue shifted, resulting in the blue-shifted A-BIC. On the other hand, the more contribution from the delocalization leads to decreasing θ ₀ for the A-BIC, that is θ ₀ being closer to 0°. If the lattice period increases, the resonance is red-shifted, resulting in red-shifted A-BIC. On the other hand, the more contribution from the delocalization also leads to decreasing θ ₀ for the A-BIC. In other words, with the characteristics of delocalization, we explained the tuning of the spectral and angular position for the A-BIC, as well as the transition of the A-BIC from the off-Γ-BIC to the at-Γ-BIC through varying the nanodisk’s size or the lattice period.