Dynamically tunable robust ultrahigh-Q merging bound states in the continuum in phase-change materials metasurface

3.1 Moving of BICs by translating the phase state of PCMs

To better understand the underlying physics, we investigate the merging process of PCs by conducting mode analysis. Figure 2(a) depicts the calculated energy bands, where the TM (transverse magnetic) mode of TM1 energy band (red curve) includes symmetry-protected BICs at the Γ -point and accidental BICs at each highly symmetry axis, while the second mode TM2 energy band has only symmetry-protected BICs located at the Γ -point, and thus we focus on the TM1 energy band. The proximity of the accidental BICs to the Γ -point is advantageous for subsequent tuning. Figure 2(b) illustrates the electric field distribution within the band at the Γ -point, highlighting the predominance of the E _z components, which is characteristic of TM-like modes. We have investigated the merging process of this energy band along the X-Γ-M direction by transforming the state of Sb₂S₃ from amorphous to crystalline, corresponding to different curve-colors (Figure 1(c)). When Sb₂S₃ is in the crystalline state, the separated accidental BICs located at k   =   0.03 a 2 π migrate towards the Γ -point to form a merging BIC. The change in Q-factor before and after merging can be characterized by scaling rules [14], as shown in Figure 2(d). The result illustrates the fitting of Q-factor in two distinct scenarios; it is evident that for the isolated accidental BICs (depicted in black-line with hollow circles), the geometric decay of Q-factor follows Q ∝ 1 k k + k accidental k − k accidental 2 [29]. Upon transition of Sb₂S₃ to its crystalline state, the decay of Q-factor for the merging BIC becomes Q ∝ 1 k 6 (depicted in red-line with hollow circles), significantly lower than that for isolated accidental BICs.

Figure 2:

Energy band and Q-factor of the metasurface. (a) Calculated band structure. The specific BICs in energy band is highlighted with dots and arrows. (b) The electric field z-component profile (Z=0) for one unit at Γ-point for the crystalline state PCMs. (c) Calculated Q-factor values for Sb₂S₃ in its amorphous (black) and crystalline (red) states. Upon transformation to the crystalline state (red), a merging BIC is formed, leading to high Q-factor achieved over a wider range of spatial frequencies. (d) The Q-factor scaling rules exhibit changes before (black) and after (red) the merge of the BICs, varying as a function of the wave vector along the Γ-M direction.

Based on the multipole decomposition analysis of the Cartesian coordinate system, we have investigated the components of the individual modes of the TM1 band, as shown in Figure 3(a). The toroidal dipole mode is dominant due to its weak coupling to incident light and relatively large Q-factor, which provides better merging of the BICs scaling rules in k-space [30]. This can be observed from the results of the electric (E) and magnetic (H) fields distribution shown in Figure 3(b and c), where the toroidal dipole mode with a circular electric field and corresponding toroidal magnetic field is seen.

Figure 3:

Multipole expansion and mode profile. (a) Multipole decomposition analysis reveals that the selected bands are predominantly influenced by a toroidal dipole mode. The abbreviations ED, MD, TD, EQ and MQ denote electric dipole, magnetic dipole, toroidal dipole, electric quadrupole, and magnetic quadrupole, respectively (b–c) Mode profiles, along with the corresponding electric field vectors (left) and magnetic field vectors (right), demonstrating the presence of the dipole mode in the x-y cross section.

The tunability of the resonant mode is attributed to the interface discontinuity of the electric field, elucidating the physical mechanism underlying this phenomenon [31]. The TM-like electromagnetic field can be given by ( H _x , H _y , E _z ) that in our case. We can obtain the coupling equations of different channels from Maxwell’s equations [32].

Here, β 0   =   2 π a , m and n are the wave orders in the xy plane, and m’ and n’ denote the source wave orders [33]. By the Fourier transform, 1 ε r is expanded into 1 ε r   =   κ a   +   ∑ κ mn e − im β 0 x − in β 0 y and 1 ε r   =   1 ε l ≜ κ b inside and outside the PCs, respectively. The guided mode resonance depends on the guided mode (β) and Bloch modes ( β mn   =   m β 0 x ̂   +   n β 0 y ̂ ) at different orders (m, n), where β   =   β mn . Owing to geometrical symmetry, all coupling coefficients (κ _mn) are symmetric at the point Γ = 0, resulting in complete cancelling interference. The intensity amplitudes of the in-plane and surface couplings are comparable yet opposite in sign, which effectively nullifies the total radiation (approaching zero) and consequently gives rise to stationary at-Γ BICs. For the accidental BICs in the Γ − X and Γ − M direction, the wave vectors form a new triangular symmetry, achieving a weighted destructive interference and the overall radiation is suppressed (close to zero). By continuously modulating the wave vectors, such weighted destructive interference always occurs with the new symmetry, resulting in tunable light trapping. Therefore, when the permittivity of the PCMs varies with the phase transition state, the out-of-plane profile of the individual channels change accordingly, modifying the coupling weights, which leads to the location shifts of the tunable BICs and the merging of the BICs. For further details and comprehensive information, please refer to the supplementary materials provided. Moreover, Δκ ≜ κb − κa, representing the disparity between the coefficients of the Fourier transforms of the permittivity outside and inside the PCs, while δ d ≜ δ z − d − δ z + d denotes the surface coupling strength in TM-like mode. It is clear that δ d and ▵ κ contribute to the resonant modes both inside and outside the PCs. However, δ d is so weak that it barely influences the electric field of eigenmodes in the out-of-plane direction, given the extremely thin (10 nm) thickness of PCMs film [31].

The Q-factor range and topological nature of the BICs are effectively illustrated through contour plots and far-field polarization maps, as shown in Figure 4, with white arrows indicating the direction of polarization. Such BICs are vortex centers in the polarization direction of the far-field radiation, carrying a conserved and quantized topological charge defined as the number of times the polarization vector encircles the BICs [13]:

where c k   =   c x k x ̂   +   c y k y ̂ represents the spatially averaged projection of the electric field onto a cell at the far-field level, pointing in the direction of polarization of the resonance in the far-field, and is referred to as the ‘polarization vector’. φ(k) = arg [c _x (k) + ic _y (k)] denotes the angle of the polarization vector. C in the integral represents the closed-loop path around the BIC in k -space along the counterclockwise direction. When Sb₂S₃ is in the amorphous state, the distribution of accidental BICs exhibits a rhombic pattern. The topological charge of the central BIC and those along the x-axes and y-axes is +1, while the remaining BICs have a topological charge of −1, see result in Figure 4(a). Upon transitioning Sb₂S₃ to its crystalline state, the isolated BICs positioned along the high-symmetry axes converge at the central Γ -point, coalescing into a single merging BICs. This merging process leads to the mutual annihilation of accidental BICs with opposing topological charges of +1 and −1, yielding a merging BICs with a net topological charge of +1. Consequently, the momentum space exhibits an expanded spectrum of wave vectors endowed with elevated Q-factors, as illustrated in Figure 4(b). More detailed information can be found in the Supplementary Information. This implies that the merging BICs is robust to external perturbations and holds potential for significant advancements in directional lasing, as well as manipulation of vortex beams [34].

Figure 4:

Merging multiple BICs by manipulating the phase state of PCMs. Distribution of the Q-factors calculated when the PCMs is in the amorphous (a) and crystalline (b) states, the topological charge of each BICs is marked on the plot. The polarization vectors around the BICs are indicated by the white-arrows.

3.2 Changing the shape of voids and manufacturing defect robustness

To verify the universality of the merging BIC, we modified the geometry of the holes in the etched PCs structure, as depicted in Figure 5. Specifically, square lattices with circular (a), rhombic (b), and ortho-hexagonal (c) holes were fabricated. Despite these changes, both mirror symmetry and C 4 z symmetry of the PCs remains unchanged, thereby leaving the merging of BICs unaffected. The results of variations in Q-factor of the merging BICs under these three different hole shapes are presented in Figure 5(d–f), with parameters h = 670 nm, 655 nm, and 685 nm while keeping other variables unchanged. One can see that accidental BICs are still able to form merging BICs as the phase state of Sb₂S₃ changes, irrespective of the morphology of the holes.

Figure 5:

Simulated Q-factors of different unit-cell topology. The magnitude of the Q-factors (d–f) before and after the phase transition were calculated separately for square lattices with circular (a), rhombic (b) and ortho-hexagonal (c) holes. The amorphous state is indicated by the black-solid-line, the crystalline state by the red-solid-line, and the structural dimensions are labeled on the graphs.

Based on the results in Figure 5, it is evident that the merging BIC has the capability to achieve a higher Q-factor across a broader range of wave vectors. This characteristic enables the suppression of out-of-plane scattering losses resulting from inevitable fabrication defects. Here, we examine the stability of this structure against three types of manufacturing errors. (1) The first scenario involves a deviation in the length of the etched holes from the design value, as shown in Figure 6(a). In this instance, the symmetry remains intact and the merging case continues to exists. (2) The second scenario arises when the originally designed square holes become distorted into parallelogram-shaped apertures (Figure 6(b)), yet this distortion does not compromise the merging of BICs. (3) Subsequently, we examined a case where the square holes are substituted with prismatic-conical concave surfaces featuring varying lengths at their top and bottom ends (Figure 6(c)). In this instance, the mirror symmetry is disrupted, rendering it unfeasible to simultaneously suppress radiation in both vertical directions, consequently leading to a decrease in the Q-factor [35]. However, in all three cases, the Q-factor values of the merging BICs exhibit a significantly larger magnitude compared to those of the accidental BICs across a broader range of wave vectors. Consequently, the merging BICs demonstrate an ability to maintain a high Q-factor in the presence of manufacturing defects.

Figure 6:

Robustness of the designed structures under different manufacturing errors. The Q-factor values for the phase transition from amorphous (first-row) to crystalline (second-row) are calculated under various conditions: when the length of the etched holes deviates from the design value (a), when the etched holes are deformed into parallelogram shapes (b), and for prismatic concave holes with different lengths at the top and bottom (c). The schematic representation of a square hole shape is shown in the upper left corner, where Δb represents the degree of length change, which is 2.5 nm, 6.4 nm, and 5 nm, respectively.

The thickness of the PCMs layer as well as the thickness difference also have an important influence on the merging BICs. By calculating the change of Q value in k-space under different thicknesses (8 nm–14 nm), it is found that the off-Γ BICs gradually move toward the Γ point with the increase of thickness, and complete the merger at t = 10 nm, producing the maximum Q values. When t is further increased, the merging BICs gradually undergo annihilation. This variation implies that the merging BICs exhibits a significant degree of robustness against variations in the thickness of PCMs layer. On the other hand, when the thickness difference of PCMs is Δt=1 nm, due to the destruction of the mirror symmetry of the structure, only a symmetry-protected BICs at the Γ point is maintained. The accidental BICs degenerate into a quasi-BICs. When PCMs changes from amorphous to crystalline, the quasi-BICs move towards the Γ point. This finding suggests that the breaking of the symmetry induced by the thickness differences can precipitate the emergence of accidental BICs, thereby influencing the formation of merging BICs (additional details and comprehensive information can be found in the supplementary materials).

3.3 Inversion of topological charge

When the square holes of each cell were further twisted along the x-axis to become parallelogram holes with increased distortion, as shown in Figure 7(a), the period and thickness remained consistent at 600 nm and 640 nm, respectively, while maintaining the PCMs thickness of 10 nm. At this point, the in-plane symmetry was reduced from C 2 z to C 4 z . All incidental BICs in the momentum space exhibited clockwise shifts compared to those in the square-hole case (Figure 7(b)). In this case, the BICs exhibit distinct displacements along various high symmetry axes, resulting in an asymmetric merging process. When changing the phase state of the PCMs, certain BICs initially merge in momentum space along the diagonal direction, ultimately giving rise to the emergence of a novel singular BICs at the Γ -point with its topological charge reversed from −1 to +1. Therefore, despite the nonlinearity of the polarization state, it leads to the rotation of radiated light around the polarization direction of the BICs in momentum space [36]. Additional information can be found in the Supplementary information. Furthermore, in momentum space, the intrinsically entangled topology surrounding the BICs gives rise to an additional PB (Pancharatnam-Berry) phase that enables the emission of vortex light. The reversal of the topological charge holds significant potential for applications aimed at manipulating the polarization state of vortex beams near the Γ -point [37], [38]. More detailed information can be found in Supplementary Information.

Figure 7:

Calculated Q-factor values and polarization vectors for severe topology (square) distorted parallelogram hole. (a) Schematic diagram of the designed structure, where c = 0.1b. The Q-factor values are presented for the amorphous (b) and crystalline (c) state of the PCM, respectively, with the topological charge of the BICs marked on the plot. The white arrows denote the polarization vectors of the BICs.