Merging bound states in the continuum by harnessing higher-order topological charges

Bound states in the continuum (BICs) can confine light with a theoretically infinite Q factor. However, in practical on-chip resonators, scattering loss caused by inevitable fabrication imperfection leads to finite Q factors due to the coupling of BICs with nearby radiative states. Merging multiple BICs can improve the robustness of BICs against fabrication imperfection by improving the Q factors of nearby states over a broad wavevector range. To date, the studies of merging BICs have been limited to fundamental BICs with topological charges ±1. Here we show the unique advantages of higher-order BICs (those with higher-order topological charges) in constructing merging BICs. Merging multiple BICs with a higher-order BIC can further improve the Q factors compared with those involving only fundamental BICs. In addition, higher-order BICs offer great flexibility in realizing steerable off-Γ merging BICs. A higher-order BIC at Γ can split into a few off-Γ fundamental BICs by reducing the system symmetry. The split BICs can then be tuned to merge with another BIC, e.g., an accidental BIC, at an off-Γ point. When the in-plane mirror symmetry is further broken, merging BICs become steerable in the reciprocal space. Merging BICs provide a paradigm to achieve robust ultrahigh-Q resonances, which are important in enhancing nonlinear and quantum effects and improving the performance of optoelectronic devices.

C . As can be seen from Table S1, for m z C symmetry, the charges of the polarization vortex can be determined up to the uncertainty of n m × , where n is an integer which can be nonzero for higher bands 1,3 . For the discussion in the main text, the TE1 band at Γ belongs to the B representation of 6v C symmetry with allowed charge -2. When the symmetry is reduced from 6v C to 2v C , the TE1 band at Γ belongs to the B representation of 2v C , thus the allowed charge are 0 or -2 which correspond to the left and middle panels of Fig. 2(c), respectively. Meanwhile, since a topological charge cannot suddenly disappear, a higher charge should split into multiple integer charges or half-integer charges 4

II. Theoretical models for polarization vortex and scaling rules of Q factor
Since the polarization vector, ( ) ( ) ( ) x y c ic φ = + k k k , of the far-field radiation is a continuous function of k in the reciprocal space, we can expand them into Taylor series around the Γ point. Then we can derive the form of polarization vortexes by implementing the symmetry constraints to determine the coefficients of the Taylor series 5 .
Taylor series can be written as: m n x mn mn c a k k Where x c and y c are both real, , where the additional " / + − " comes from the symmetry/antisymmetry representation of the plane wave basis 1 .
As an example, we consider a PCS that belongs to the 6v C point group and the at-Γ eigenmode has the B1 representation. For the B1 representation of the 6v C point group, which gives rise to The polarization vortexes of other representations can be obtained similarilty. In this way, we can obtain polarization vectors radiated from the PCS with point groups of 3v C , 4v C and 6v C , as summarized in Table 2. Polarization vortexes in the momentum space are shown in Fig. S1, where BICs are located at the center of the polarization vortexes, here the Γ point. The eigenvalues of the rotation and mirror symmetry on polarization vortexes are the same as that of the corresponding eigenvector. The topological charge of a vortex is defined as the winding number of the polarization direction counterclockwise enclosing the vortex center. As shown in Fig. S1, there is a , and a higher-order BIC with a charge q = −2 of ( ) ( ) Topological charges are also provided at the centers of the vortexes.
The geometrical decay of the Q factor away from a BIC follows a scaling rule that can be derived from . Therefore for isolated BICs at the Γ point, the scaling rule is for a BIC with a charge 1 q = ± , and it becomes 4 1 Q k  for a BIC with a charge 2 q = − . More general, the scaling rule of an isolated BIC with a charge q n = ± is We can see that the Q factor in the vicinity of an isolated BIC can be magnificently enhanced by increasing the topological charge of the BIC. Meanwhile, when another accidental BIC appears close to the isolated BIC at Γ, the scaling rule is modified. Since the accidental BICs considered here all exhibit topological charge 1 ± , the scaling rule in the vicinity of each accidental BIC is

III. Evolution of band structures
We consider the evolution of band structures when the system symmetry is reduced by elliptic cylindrical holes.

IV. Evolution of BICs with the variation of structural parameters
The evolution of BICs with the variation of thickness for the structure with cylindrical holes is shown in Fig.   S3. The trajectories of BICs in momentum space become visible to the eye when we plot a brightness map of Qfactors. We see that the symmetry-protected BIC always exists at the Γ point, and accidental BICs are gradually tuned to the Γ point with the decrease of thickness from t = 340nm. The merging BIC is formed at t = 324.2 nm when accidental BICs are gathered at the Γ point, as shown in Fig. 1c the main text. Subsequently, accidental BICs disappear because of topological charge annihilation. The evolution of BICs with the variation of the aspect ratio of the elliptic cylindrical hole is shown in Fig. S4.
When the semi-major axis is located along the x-axis, the symmetry-protected BIC at the Γ point is split into two BICs in the ΓΜ direction, as shown in Fig. S4. Here we can only see one BIC near Γ as we only show the positive half of the ΓΜ direction. We fix the length of the semi-minor axis at r2 = 80 nm and increase the length of the semimajor axis so as to change the aspect ratio. We start with t = 342 nm as shown in Fig. S4a. With the increase of the semi-major axis, the split BIC is gradually tuned away from the Γ point first and then returns to the Γ point. After the two split BICs merge at the Γ point, they split once again and then shift away along the ΓΚ direction. Meanwhile, accidental BICs appear in both the ΓΜ and ΓΚ directions and are tuned to shift away from the Γ point with the increase of the semi-major axis. When the thickness is decreased to t = 334 nm, as shown in Fig. S4b, the split BIC can be tuned to merge with another accidental BIC along the ΓΜ direction. When the thickness is further decreased to t = 330 nm as shown in Fig. S4c, with the increase of the aspect ratio, the split BIC and the accidental BIC gradually approach each other to form a merging BIC and subsequently annihilate with each other. When the aspect ratio is further increased, two BICs emerge and are tuned to shift away from each other. the period is a = 336 nm, the semi-minor axis is kept at r2 = 80 nm and the semi-major axis along the x-axis (r1) is varied to change the aspect ratio.
Next, we study the evolution of the split BICs and accidental BICs with the variation of thickness t. When the system symmetry is reduced from 6v C to 2v C , a symmetry-protected higher-order BIC with topological charge -2 splits into two BICs in the ΓΚ direction at t = 350 nm. As shown in Fig. S5, with the decrease of t, the two split

V. Robustness of merging BICs against fabrication imperfections
We consider the influence of three typical types of fabrication errors on Q factors. In the first type, the etched hole  Here apart from the varying parameters, other parameters are the same as Fig. 1 in the main text, t = 324.2 nm for the merging BIC and t = 300 nm for the isolated BIC.

VI. Intrinsic loss of materials
The total Q factor is determined by the contribution of both radiation loss and intrinsic loss, following  and an off-Γ point (c). Im(n) = 6 10 − in b and c. Other structural parameters keep the same as Fig. 1 and Fig. 3 in the main text for t = 324 nm and t = 334 nm, respectively.

Merging BICs in degenerate bands
As shown in Fig. 1b, TE2 Fig. 1c, Fig. 2c and Fig. 4c in the main text, respectively. The magnitude of S3 is small (< 0.04) within the interesting range which shows that the polarization vectors are almost linear.