Title: Complex Band Structures and Bound States in the Continuum: A Unified Theoretical Framework Show Chinese title

Title: 复杂能带结构和连续谱中的束缚态：统一的理论框架

Abstract: Complex band structures describe resonant modes in periodic systems finite in one direction, crucial for applications in photonics like sensors and lasers. The imaginary frequency component, $\omega''$, tied to quality factors, is often probed through numerical or phenomenological models, limiting deeper insights. Here, we introduce a first-principles framework deriving complex band structures from scattering matrix poles, integrated with perturbation theory to define minimal Hilbert spaces based on interacting Bloch waves. Key results include a two-band model yielding $\omega'' = C(\mathbf{k}_{||})\delta^2$ and identifying accidental bound states in the continuum (BICs) at $C(\mathbf{k}_{||})=0$, with dual Fabry--P\'erot modes from impedance degeneracy; a three-band model reveals Friedrich--Wintgen and symmetry-protected BICs plus linewidth behaviors; inclusion of orthogonal polarizations characterizes far-field states and exceptional points; and the above model can be extended to two-dimensional systems. This unified approach advances light confinement studies, including nano-photonics and interdisciplinary wave physics with broad implications for high-performance optical devices. Show Chinese abstract

Abstract: 复杂能带结构描述了一维有限周期系统中的共振模式，对于光子学中的传感器和激光器等应用至关重要。 虚频分量$\omega''$与品质因数相关，通常通过数值或现象模型进行探测，限制了更深入的理解。 在此，我们引入一种从散射矩阵极点推导复数能带结构的第一性原理框架，并结合微扰理论，基于相互作用的布洛赫波定义最小希尔伯特空间。 关键结果包括一个两带模型产生$\omega'' = C(\mathbf{k}_{||})\delta^2$并在$C(\mathbf{k}_{||})=0$处识别出偶然的连续谱束缚态（BICs），由阻抗简并产生的双Fabry-Pérot模式；一个三带模型揭示了Friedrich-Wintgen和对称性保护的BICs以及线宽行为；正交极化包含可表征远场态和例外点；上述模型可以扩展到二维系统。 这种统一的方法推进了光约束研究，包括纳米光子学和跨学科波动物理，对高性能光学器件具有广泛影响。