Achieving asymmetry parameter-insensitive resonant modes through relative shift–induced quasi-bound states in the continuum

2.1 Design principle for the proposed DSMs

Figure 1 shows the schematic diagram of the proposed DSMs and its band properties. As shown in Figure 1(b) and (c), the supercell of the DSMs consists of two Si disks supported by silica (SiO₂) substrate. The Si meta-dimers in the supercell have the same height h, and their widths are l ₁ and l ₂, respectively. The periods of the supercell of the DSMs in the x and y directions are P _x and P _y , respectively. The refractive indexes of Si and SiO₂ are 3.48 and 1.47, respectively. As the two Si disks are located in the center of half area of the supercell, the distance between them along the x direction is D. The shift of a Si disk from its center along the and x direction is marked as Δ, and the structural asymmetry parameter of the DSMs can be described by the relative shift, which is defined as δ = Δ/D. In the case of l ₁ = l ₂ and δ = 0, the DSMs are degenerated into the simple lattice structure where its unit cell consists of a single Si disk, and there is no band structure in the wavelength region of interest for the structural parameters indicated in Figure 1. However, by introducing the relative shift with δ ≠ 0, the supercell is formed and the band structure of the DSMs can be realizable due to the in-plane inversion (C₂) symmetry breaking as well as the Brillouin zone folding along the x direction [31], [32], [33]. Therefore, it is possible to excite the previously inaccessible modes (dark modes) by free-space illumination through the relative shift of the supercell of the DSMs.

Figure 1:

Schematic diagram of the proposed DSMs and its band properties. (a) Schematic diagram of the DSMs under the plane wave illumination, the supercell of the DSMs is indicated by the green dash line. (b) Schematic diagram of the supercell of the DSMs. (c) Vertical view of the supercell of the DSMs in the xy plane, where the shift of a Si disk from its center is indicated as Δ. (d) Band diagram of the DSMs along X′ − Γ − X. The parameters are P _x = 2P _y = 1000 nm, l ₁ = l ₂ = 350 nm, h = 220 nm, and δ = Δ/D = 0.34. (e) Electromagnetic field distributions of the DSMs at Γ point for the TM-like and TE-like modes. TM#1_B and TM#2_A are z component of electric-field E _z overlaid with the in-plane magnetic-field vector H _xy ; TE#1_B and TE#2_A are z component of magnetic-field H _z overlaid with the in-plane electric-field vector E _xy .

Figure 1(d) shows the band structure of the DSMs with δ = 0.34 along the X′ − Γ − X. The dispersion curve of the DSMs shows TM-like mode (with strong E _z component and negligible E _x , E _y , and H _z components) and TE-like mode (with strong H _z component and negligible H _x , H _y , and E _z components). These TM-like and TE-like modes shows bonding (field highly confined in Si disks) and antibonding (field mainly trapped by air gaps) features, which are indicated as the subscript of B and A, respectively. That is, TM#1_B (TE#1_B) indicates the TM (TE) bonding modes, and TM#2_A (TE#2_A) indicates the TM (TE) antibonding modes. Figure 1(e) shows electromagnetic field distributions of the DSMs at Γ point for the TM-like and TE-like modes. As can be seen in Figure 1(e), the odd–even symmetry of the eigenmodes of the DSMs are different. The TM#1_B (TE#1_B) modes have odd-like transverse electric-field (magnetic-filed) profiles, while the TM#2_A (TE#2_A) modes have even-like transverse electric-field (magnetic-filed) profiles. Basically, these two sets of modes with the opposite parity are related to the electromagnetic duality, which reflects the symmetry of Maxwell’s equations with respect to the electric and magnetic components of electromagnetic waves [5].

Figure 2 depicts two-dimensional (2D) reflection maps of the DSMs as functions of structural asymmetry parameters, considering normally incident plane wave illumination. All other parameters remain consistent with those presented in Figure 1. We evaluate two distinct categories of asymmetry parameters: the conventional parameter α, linked to in-plane inversion symmetry breaking in DSMs, and the relatively less-explored parameter δ, associated with mirror symmetry breaking within DSMs. Upon close examination of Figure 2, it becomes evident that for both δ = 0 and α = 0, no resonances manifest in either scenario, owing to the inherent simplicity of the lattice structure in DSMs. However, these initially nonresonant dark modes undergo a transformation into high-Q radiative modes as α or δ deviate from zero due to structural symmetry perturbation. It is noteworthy that the linewidths of these high-Q modes expand with increasing absolute values of α or δ. Of particular interest, when δ ≠ 0 and α ≠ 0 for DSMs, high-Q modes associated with both mirror symmetry breaking and in-plane inversion symmetry breaking can be excited simultaneously (see Supplementary S1, Supporting Information).

Figure 2:

Reflection 2D maps of the DSMs as functions of the asymmetry parameters under normally incident plane waves, the schematic diagrams of the supercell of the DSMs are inserted in the figures; other parameters are the same as Figure 1. (a) and (b) are reflection 2D maps of the DSMs with l ₁ = l ₂ as functions of δ for the x and y polarizations, respectively. (c) and (d) are reflection 2D maps of the DSMs with δ = 0 as functions of α for the x and y polarizations, respectively, where α = (l ₁ − l ₂)/l ₂.

Specifically, Figure 2(a) and (b) shows reflection 2D maps of the DSMs with l ₁ = l ₂ as functions of δ for two orthogonal polarization states. In these figures, it can be seen that two QBICs modes can be excited with δ ≠ 0, which are associated with the eigenmodes of TM#1_B and TE#1_B for the x and y polarizations, respectively. For the x polarization, the relative shift of the DSMs leads to the in-plane inversion symmetry breaking of the magnetic field about the y–z plane. Conversely, for the y polarization, it corresponds to the in-plane inversion symmetry breaking of the electric field about the y–z plane. Consequently, two high-Q modes linked to two symmetry-protected BICs can be selectively switched under orthogonal polarization states. Figure 2(c) and (d) further illustrate that two QBICs modes can also be excited as the C₂ symmetry is disrupted with α ≠ 0. These modes are associated with the eigenmodes of TM#2_A and TE#2_A for the x and y polarizations, respectively. It is noteworthy that in the case of x polarization in Figure 2(c), the high-Q mode of the DSMs tends to vanish as α falls below −0.3. This behavior arises from the fact that the (±1, 0) diffraction orders propagate in the substrate near the cutoff wavelength of 1470 nm, corresponding to the Rayleigh anomaly [34]. As α varies, it redistributes the field energy of the antibonding mode confined in the air gap, resulting in significant shift of the location of the high-Q modes. In contrast, the location of the high-Q modes is robust to the variation of δ due to their bonding mode characteristics, where field energy is well confined in the Si meta-dimers thus insensitive to the variation of δ. This property proves advantageous for the robust excitation of high-Q modes at specific wavelengths. In forthcoming studies, our focus will center on the optical performance of DSMs, particularly with regard to the less-discussed asymmetry parameter of relative shift δ.

2.2 Analysis on the excitation of relative shift-induced QBICs

Figure 3 shows optical performances of the DSMs as functions of the relative shift δ, with the remaining parameters remaining consistent with those presented in Figure 1. In Figure 3(a)–(c), the Q-factor of the reflection response of the DSMs are defined as λ _r /∆λ, where λ _r is the resonant wavelength and ∆λ is its linewidth. According to the perturbation theory [20], the radiative quality factor Q _rad of a metasurface is inversely proportional to the inverse radiation lifetime γ _rad, and it shows an inverse square dependence on the structural asymmetry parameter as long as the QBIC frequency is below the diffraction limit of the substrate. Therefore, the Q-factor of the DSMs related to the relative shift δ can be fitted as:

where η is a proportionality constant, c is the speed of light in vacuum, S is the area of the supercell, and ω is the angular frequency. As illustrated in Figure 3(a)–(c), divergent radiative Q-factor that tends to infinity is occurred at δ = 0 for both the x and y polarizations, indicting the existence of the BICs. However, even a minimal relative shift leads to a significant reduction in the Q-factor of the DSMs. The fitting results for the Q-factor align closely with those obtained from simulations, for both x and y polarizations. Specifically, for small values of δ, the behavior of the DSMs’ Q-factor exhibits a clear inverse quadratic trend. Slight discrepancies emerge at larger δ, primarily due to deviations from the slight perturbation approximation.

Figure 3:

Optical performances of the DSMs as functions of the relative shift δ. (a) and (b) are Q-factor of the DSMs as function of δ for the x and y polarizations, respectively. (c) and (d) are the eigenmode responses of the DSMs as function of δ for the x and y polarizations, respectively; the error bars indicate the magnitude of the mode inverse radiation lifetime. Other parameters are the same as Figure 1.

Figure 3(b)–(d) shows the eigenmode responses of the DSMs as function of relative shift δ for the x and y polarizations, respectively. The complex eigenfrequency of the eigenmode containing real and imagery parts can be written as ω ~ = Re( ω ~ )+Im( ω ~ ), where Re( ω ~ ) corresponds the resonance location, and 2·Im( ω ~ ) corresponds the mode inverse radiation lifetime. As can be seen in Figure 3(b)–(d), the resonance locations of the eigenmodes are insensitive to the variations of the relative shift δ, while the radiation leakage related to the mode inverse radiation lifetime can be gradually enhanced with the increase of |δ|, which are in accord with the reflection features of the DSMs shown in Figure 2(a) and (b). Therefore, controllable radiation leakage operating at the desired wavelength is feasible through the relative shift of the Si meta-dimers.

Figure 4 shows optical properties of the DSMs with δ = 0.34. All other parameters remain consistent with those presented in Figure 1. As the relative shift of the Si meta-dimers is associated with the interference between the discrete resonant mode and the broadband incident waves, the relative shift–induced QBICs can be described by the typical Fano formula [35]:

where ω ₀ is the resonant frequency, R ₀ is the reflection offset, κ is the resonance linewidth, A ₀ is coupling constant between the discrete and continuum states, and q is the Breit–Wigner–Fano parameter determining asymmetry of the resonance profile.

Figure 4:

Optical properties of the DSMs with δ = 0.34; other parameters are the same as Figure 1. (a) and (e) are reflection responses of the DSMs for the x and y polarizations, respectively. (b) and (f) are the far-field scattering power for the x and y polarizations, respectively. (c) and (d) are near-field distribution at reflection peak of 1472.6 nm for the x polarization. (g) and (h) are near-field distribution at reflection peak of 1540.0 nm for the y polarization. Specifically, colors in (c) and (d) indicate the distributions of y component of magnetic-field H _y and z component of electric-field E _z , respectively; black arrows in (c) and (d) indicate displacement current vector and magnetic-field vector, respectively. Colors in (g) and (h) indicate the distributions of y component of electric-field E _y and z component of magnetic-field H _z , respectively; black arrows in (g) and (h) indicate magnetic-field vector and displacement current vector, respectively.

Figure 4(a)–(e) shows reflection responses of the DSMs for the x and y polarizations, respectively. As can be seen in Figure 4(a)–(e), the fitting results of the analytic Fano formula are in good agreement with the reflection responses for both the x and y polarizations, validating the excitation of the QBIC-induced Fano resonance accompanied with the relative shift of the Si meta-dimers. To further understand the resonant mechanism of the relative shift–induced QBICs, the far-field scatterings and the near-field distributions of the DSMs for both the x and y polarizations are investigated. In calculation, the induced current density in the supercell is extracted to perform the multipole decomposition in Cartesian coordinate [36], [37], and the far-field scatterings of the DSMs are decomposed into five major components: electric dipole (ED), magnetic dipole (MD), toroidal dipole (TD), electric quadrupole (EQ), and and magnetic quadrupole (MQ) (see Supplementary S2, Supporting Information).

As shown in Figure 4(b)–(f), for both the x and y polarizations, it can be seen that the relative shift–induced QBICs can be regarded as the TD mode as the TD resonance provides the major contributions to the far-field scattering power among the multipoles. For the x polarization at reflection peak of 1472.6 nm, as shown in Figure 4(c) and (d), the resonance feature is obvious as the magnetic field and electric field are highly trapped by the supercell of the DSMs. In particular, as shown in Figure 4(c), the magnetic field is strongly enhanced and the displacement current forms two reversed loops between the center and the edge of the Si meta-dimers at the x–z plane, indicating the out-of-plane TD mode along the z axis [38]; the distribution of the magnetic-field vector with reverse direction along the y axis in Figure 4(d) also reveals the feature of the out-of-plane TD mode. For the y polarization at reflection peak of 1540.0 nm, as shown in Figure 4(g) and (h), the field pattern also exhibits the resonance properties due to its highly localized distribution. Specifically, as shown in Figure 4(g), the magnetic field vector’s distribution forms a clockwise loop in the center of the supercell at the x–z plane, signifying the presence of the in-plane TD mode along the y-axis. The reversed loops of the displacement current vector at the x–y plane in Figure 4(h) also align with the features of the in-plane TD mode. The congruence between the far-field scatterings and the near-field distributions of the DSMs confirms that the out-of-plane and in-plane TD modes are responsible for the excitations of the relative shift–induced QBICs for the x and y polarizations, respectively.

2.3 Experimental observations of asymmetry parameter-insensitive resonant modes

Subsequently, we conduct experiments to validate the existence of relative shift–induced QBICs within the DSMs, where the resonance locations remain unaffected by variations in structural asymmetry parameters. In the experiment, the DSMs are fabricated by resist spin-coating, electron beam lithography (EBL), and inductively coupled plasma (ICP) etching techniques (see Supplementary S3, Supporting Information). Six types of samples with different δ are fabricated on a silicon-on-insulator (SOI) wafer. It’s worth noting that DSMs with 0 < δ < 0.5 are not fabricated because their resonance linewidths are too narrow to be detected in our measurement system. Scanning electron microscopy (SEM) images of the fabricated samples, displayed in Figure 5(a), reveal relatively well-matched structural parameters in alignment with the design. However, it’s important to note that, in comparison to DSMs supported by SiO₂ substrates, the resonance locations of the SOI-based structures remain insensitive to variations in δ, but their linewidths are broadened and their Q-factors significantly reduced due to mode leakage into the Si substrate via open diffraction channels (see Supplementary S4, Supporting Information).

Figure 5:

Experimental results of the DSMs. (a) SEM images of the fabricated DSMs with different relative shifts. (b) and (c) are the measured reflection responses of the DSMs for the x and y polarizations, respectively. (d) Simulated and measured resonance wavelengths of the DSMs as functions of the relative shifts of the Si meta-dimers.

Figure 5(b) presents the measured reflection responses for the x polarization. As observed, at δ = 0, there exists only a low-reflection background. However, a relative shift of the Si meta-dimers induces a resonance above this background in the short-wavelength region. The linewidth of this resonance broadens with increasing δ, yet its location remains robust against variations in δ. It’s worth noting that the peak reflection is reduced, and the resonance linewidths are broadened in the measured responses, which can be attributed to material losses, surface roughness, and finite lateral dimensions of the samples [39]. In Figure 5(c), we can observe a relative shift-induced QBIC in the measured responses in the longer wavelength region, whose location remains insensitive to δ and can also have its linewidth tuned by adjusting the asymmetry parameters. Figure 5(d) demonstrates that the measured resonance wavelengths of the DSMs exhibit a tendency in line with the simulation results, albeit with a slight redshift in the resonance location. These deviations may arise from minor variations in parameters, such as refractive indices of materials or imperfections in the surface profile of the Si meta-dimers.