Structural perturbation theory for bound states in the continuum via bifurcation of zeros and extrema of the dispersion relation

Abstract

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.