连续谱中束缚态的局部鲁棒性：通过散射矩阵本征向量延拓
Local Robustness of Bound States in the Continuum through Scattering-Matrix Eigenvector Continuation
AI总结：
本文通过散射矩阵本征向量延拓和隐函数定理，证明了简单连续谱中束缚态在参数扰动下连续变形为传播场，并引入映射度概念给出了其局部鲁棒性的拓扑解释及数值判据。
摘要：
我们考虑由亥姆霍兹方程控制的周期性结构对时间谐波平面波的衍射。连续谱中的束缚态（BICs）是准周期场，在一个周期内保持$L^{2}$有界，且出现在连续谱中嵌入的频率处。破坏BIC的扰动可导致超强共振，从而在光子学中实现各种应用。利用隐函数定理，我们展示了当系统参数在邻域内变化时，简单的BIC如何连续变形为传播场，且频率相应调整。在此设定下，场的入射系数作为散射矩阵具有固定特征值的特征向量持续存在。通过引入从参数到这些系数的映射$\mathcal{P}$，$\mathcal{P}$的零点恰好对应于BIC。当这样的零点是孤立的且定义域与余定义域的维数相等时，BIC可与小邻域内$\mathcal{P}$的映射度相关联。这一视角阐明了与BIC相关的相位奇点，并为其在给定参数下的局部鲁棒性提供了通用的拓扑解释。此外，它通过计算$\mathcal{P}$的映射度，为检测和验证BIC提供了实用的数值判据。
英文摘要：
We consider the diffraction of time-harmonic plane waves by a periodic structure, governed by the Helmholtz equation. Bound states in the continuum (BICs) are quasi-periodic fields that remain $L^{2}$-bounded over one period and occur at frequencies embedded in the continuous spectrum. Perturbations that break a BIC can lead to ultra-strong resonances, enabling various applications in photonics. Employing the implicit function theorem, we demonstrate how a simple BIC continuously deforms into a propagating field as system parameters vary in a neighborhood, with the frequency adjusting accordingly. In this setting, the incident coefficients of the field persist as an eigenvector of the scattering matrix with a fixed eigenvalue. By introducing a mapping $\mathcal{P}$ from the parameters to these coefficients, the zeros of $\mathcal{P}$ correspond precisely to BICs. When such a zero is isolated and the dimensions of the domain and codomain coincide, the BIC can be related to the mapping degree of $\mathcal{P}$ in a small neighborhood. This perspective clarifies the phase singularity associated with BICs and provides a general topological interpretation of their local robustness with respect to the given parameters. Moreover, it yields a practical numerical criterion for detecting and verifying BICs via computation of the mapping degree of $\mathcal{P}$.
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