Super Bound States in the Continuum: Analytic Framework, Parametric Dependence, and Fast Direct Computation

In periodic structures such as photonic crystal (PhC) slabs, a bound state in the continuum (BIC) is always surrounded by resonant states with their $Q$-factor following $Q\sim 1/|{\bm \beta}-{\bm \beta}_*|^{2p}$, where ${\bm \beta}$ and ${\bm \beta}_*$ are the Bloch wavevectors of the resonant state and the BIC, respectively. Typically $p=1$, but special BICs, known as the super-BICs, have $p\geq 2$. Super-BICs can significantly enhance the $Q$-factor of nearby resonant states and reduce scattering losses due to fabrication imperfections, making them highly advantageous in practical applications. However, super-BICs, requiring the tuning of structural parameters for their realization, are generally not robust. In this work, we develop a theory to classify super-BICs, determine the minimal number $n$ of tunable structural parameters needed, and show that super-BICs form a manifold of dimension $(m-n)$ in an $m$-dimensional parameter space. We also propose a direct method for computing super-BICs in structures with different symmetry. Numerical examples demonstrate that our method is far more efficient than existing methods when $n > 1$. In addition, we study the effect of structural perturbations, focusing on the transition from super-BICs to generic BICs. Finally, we analyze a class of degenerate BICs that can be regarded as Dirac points, and show that they are the intersections of super-BICs in a relevant parameter space. Our work advances the theoretical understanding on super-BICs, and has both direct and potential applications in optical design and light-matter interactions.