Perturbation theory for resonant states near a bound state in the continuum

In this work, we develop a perturbation theory to analyze resonant states near a bound state in the continuum (BIC) in photonic crystal slabs. The theory allows us to rigorously determine the asymptotic behavior of $Q$-factor and the far-field polarization. We show that the resonant states close to a BIC can be nearly circularly polarized if the scattering matrix satisfies a certain condition. Moreover, our theory offers a novel perspective on super-BICs and provides a clear and precise condition to efficiently identify them in both symmetric and asymmetric structures without requiring a merging process. For practical applications, we find a super-BIC in a square lattice of rods on a dielectric substrate. Our theory addresses the non-Hermitian nature of the system, can be generalized to treat other structures that support BICs, and has potential applications in resonance and chiral optics.