标题： 连续区的超束缚态：解析框架、参数依赖性与快速直接计算 显示英文标题

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

摘要： 在光子晶体（PhC）薄膜等周期性结构中，连续区内的束缚态（BIC）总是被共振态所包围，其共振态的 $Q$-因子满足 $Q\sim 1/|{\bm \beta}-{\bm \beta}_*|^{2p}$的关系式，其中 ${\bm \beta}$和 ${\bm \beta}_*$分别为共振态和 BIC 的布洛赫波矢。通常情况下 $p=1$，但存在一类特殊的 BIC，称为超-BIC，其满足 $p\geq 2$。超-BIC 可以显著增强附近共振态的 $Q$-因子，并减少由于加工缺陷引起的散射损耗，在实际应用中具有明显优势。然而，超-BIC 的实现需要调节结构参数，因此通常不够稳健。 在这项工作中，我们发展了一种理论来分类超BIC（super-BIC），确定所需的可调结构参数的最小数量$n$，并表明超BIC在$m$维参数空间中形成一个维度为$(m-n)$的流形。 我们还提出了一种直接的方法，用于计算具有不同对称性的结构中的超BIC。 数值例子表明，当$n > 1$时，我们的方法比现有方法效率高得多。此外，我们研究了结构扰动的影响，重点关注从超BIC到普通BIC的转变。 最后，我们分析了一类可以看作狄拉克点的退化BIC，并表明它们是在相关参数空间中超BIC的交点。 我们的工作推进了对超BIC的理论理解，并在光学设计和光-物质相互作用中有直接和潜在的应用。 显示英文摘要

摘要： 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.