Complex Band Structure and Bound States in the Continuum: A Unified Theoretical Framework
Band structure analysis is central to understanding wave propagation in periodic media; however, it becomes challenging in open systems owing to energy leakage. Photonic crystal (PhC) slabs exemplify such systems, featuring periodicity in the $x$-$y$ plane and finite extent in the $z$-direction, and supporting diverse guided-mode resonances whose interactions give rise to phenomena such as bound states in the continuum (BICs), exceptional points (EPs), and circular polarisation states. Although numerical simulations can reveal these effects, effective non-Hermitian Hamiltonians are often employed to elucidate the underlying physical mechanisms. This approach, however, relies on manually selected resonant modes and may suffer from basis incompleteness. Here, a systematic first-principles approach is presented to derive the complex band structure. The minimal channels in the scattering matrix, either open or closed, are determined by the number of propagating bulk Bloch waves. The interactions between these waves fully reveal the complex band structure. For instance, two Bloch waves predict the leading-order imaginary frequency $\omega''$ and identify accidental BICs, each associated with a dual Fabry--P\'erot mode, whereas three waves reveal robust Friedrich--Wintgen and symmetry-protected BICs together with the associated linewidth behaviours. Orthogonally polarised waves are further incorporated to characterise far-field polarisation and EPs. When extended to a two-dimensional periodic structure, this framework accurately predicts $\omega''$, encompasses all known BICs, and tracks their evolution with system parameters. Overall, this first-principles approach provides a unified foundation for studying complex band structure and facilitates the exploration of light confinement in periodic media.