基于色散关系零点与极值分岔的连续谱束缚态结构微扰理论
Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation
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中文总结 AI 辅助
该研究提出一套微扰理论，通过色散关系零点与极值的演化规律，揭示连续谱束缚态（BIC）在各类结构微扰下的行为，还预测了超BIC在PT对称微扰下的新特性，为相关研究提供通用框架。
中文摘要 AI 辅助
在无损耗周期结构中，连续谱束缚态（BIC）对应复色散关系 $k = k(\beta)$ 的实零点与虚部的局部极大值，其中 $\beta$ 为布洛赫波数。结构微扰会使色散曲线发生形变，可能破坏、移动或分裂BIC，现有研究已验证了无损耗保对称微扰、破对称微扰及耗散微扰的相关现象。本文提出一套全面的微扰理论，重点研究各类结构微扰下 $\mathrm{Im}[k(\beta)]$ 的实零点与极值点的演化。该理论揭示了激光阈值模式（LTM）的存在：当微扰包含增益（无论是否伴随平衡损耗）时，LTM也是 $\mathrm{Im}[k(\beta)]$ 的零点。借助局部泰勒展开与Puiseux级数，本文确定了不同类型BIC在各类结构微扰下实零点与极值点的数量、位置及主导阶标度关系。该理论可复现已知结果，并预测超BIC在 $\mathcal{PT}$ 对称微扰下的新行为：具体而言，对应 $\mathrm{Im}[k(\beta)]$ 四阶零点的传播型超BIC会分裂为两个实零点，分别代表两个BIC或两个LTM；对称驻波则会分裂为两个BIC-LTM对。本文理论为研究无损耗及非厄米周期结构中的BIC及邻近共振模式提供了通用框架。
英文摘要
In a lossless periodic structure, a bound state in the continuum (BIC) corresponds to a real zero and a local maximum of the imaginary part of a complex dispersion relation $k = k(β)$, where $β$ is the Bloch wave number. A perturbation of the structure deforms the dispersion curve and may destroy, move or split the BIC, as demonstrated in existing studies involving lossless symmetry-preserving perturbations, symmetry-breaking perturbations and dissipative perturbations. We present a comprehensive perturbation theory, emphasizing the evolution of the real zeros and extreme points of $\mathrm{Im}[k(β)]$ under various structural perturbations. In particular, our theory reveals the existence of lasing threshold modes (LTMs), which are also zeros of $\mathrm{Im}[k(β)]$ when the perturbation involves gain with or without balanced loss. Using local Taylor expansions and Puiseux series, we determine the number, locations, and leading-order scaling of real zeros and extreme points for various types of BICs under different types of structural perturbations. The theory recovers known results and predicts new behavior for super-BICs under $\mathcal{PT}$-symmetric perturbations. Specifically, a propagating super-BIC corresponding to a fourth-order zero of $\mathrm{Im}[k(β)]$ splits into two real zeros representing either two BICs or two LTMs, and a symmetric standing wave splits into two BIC-LTM pairs. Our theory provides a general framework for studying BICs and nearby resonant modes in both lossless and non-Hermitian periodic structures.
