Geometry symmetry-free and higher-order optical bound states in the continuum

Geometrical symmetry plays a significant role in implementing robust, symmetry-protected, bound states in the continuum (BICs). However, this benefit is only theoretical in many cases since fabricated samples’ unavoidable imperfections may easily break the stringent geometrical requirements. Here we propose an approach by introducing the concept of geometrical-symmetry-free but symmetry-protected BICs, realized using the static-like environment induced by a zero-index metamaterial (ZIM). We find that robust BICs exist and are protected from the disordered distribution of multiple objects inside the ZIM host by its physical symmetries rather than geometrical ones. The geometric-symmetry-free BICs are robust, regardless of the objects’ external shapes and material parameters in the ZIM host. We further show theoretically and numerically that the existence of those higher-order BICs depends only on the number of objects. By practically designing a structural ZIM waveguide, the existence of BICs is numerically confirmed, as well as their independence on the presence of geometrical symmetry. Our findings provide a way of realizing higher-order BICs and link their properties to the disorder of photonic systems.


Supplementary Note 1．Eigenmode analysis of ZIM with objects
In simulation: Full-wave simulations were carried out by using software COMSOL Multiphysics to study the eigenfrequency and eigenmode profiles. In Fig. 2  proportional to the square of  (see Supplementary Fig. 3b). These results further confirm the existence of higher-order BICs.

Supplementary Note 2．Analysis of BIC in the ZIM embedded with a rectangular object and a cylindrical object with different materials
According Eq. (1) in the main text, if a rectangular object with permittivity r  and a cylinder with permittivity c  embedded in the ZIM (see Supplementary Fig. 8a), the transmission coefficient can be written as where cc kc   , c  is the permittivity of the cylinder and R is the radius of the cylinder. The magnetic flux inside the rectangular object is 1 Here r E is electric field in the rectangular object [1], given by ,,

Supplementary Note 3. Analytical derivation of Q factor of quasi-BIC resonances
To calculate the Q factor in Fig. 4c, we define it as max / Q n n  ( As mentioned in the main text and shown in Supplementary  R . In this case, the system supports (N-2)-fold degenerate BICs, then only one transmission peak in transmission spectrum is left. Next, we will derive the Q factor of this transmission peak. In this case, Supplementary Eq. (7) can be written as Substituting Supplementary Eq. (9) to Supplementary Eq. (8) and using In Supplementary Eq. (11), the right-hand side of the equation is approximately equal to . For easy analysis, we set 0 a b 1 3 2 With g1 and g2, the solutions of Supplementary Eq. (11) are Because the value of ab xx  is very small, max x can take an approximation, Finally, the Q factor is defined by , and after some simplifications, we have where   The permittivity is modeled with waveguide dispersion of TE10 [2], where b  is the relative permittivity of the medium filling in the waveguide, H is height of waveguide, and  is the working wavelength in free space. Supplementary Eq. (16) is similar to the Drude model.
The red curve in Supplementary Fig. 10 shows the relationship between eff  and frequency. Clearly, eff  changes slowly with frequency so that the ENZ window is relatively broadband. After doping a silicon rod into the waveguide junction, the effective permeability of the waveguide junction can be written as [3],

Supplementary Note 5. Geometry symmetry-free BIC induced by ENZ medium.
In epsilon near zero (ENZ) medium, the Eq. (1) in the main text can be generalized to [4], where S l w  and d S is the sum of areas of embedded objects. Likewise, we consider two cylinder objects for illustration and comparison. Supplementary Fig. 11a shows the calculated transmission vs  Supplementary Fig. 11c, d) or the right object one (see the right pattern in Supplementary Fig. 11c, d).
For the quasi-BIC, due to the term of  , the magnetic flux induced by two object are not exactly out of phase at all (see the middle pattern in Supplementary Fig. 11c, d) Fig. 2a in the main text. We can clearly see that the BIC is red-shifted and preserved with the increase of permittivity. The working frequencies of BICs are slightly shifted accordingly, but still located in the frequency window of ZIM, whose bandwidth is about 4%, as displayed in Fig. 5a. Therefore, similar to the conventional symmetry-protected BIC, the ZIM-based BIC also exists at any permittivity of dielectric voids, as long as the working frequency of the proposed BICs is located in the ZIM window.  . b

Supplementary
and c are out-of-plane magnetic field distributions at transmission peak A and peak B, respectively.