Theory of Supercritical Coupling And Generalized Bound States in the Continuum
Bound states in the continuum (BICs) arise from destructive interference suppressing radiation despite spectral overlap with the continuum. Here we show that Friedrich--Wintgen interference naturally emerges from a bright--dark supermode decomposition of resonances coupled through a shared radiation channel. In this basis, any finite leakage of a quasi-BIC induces a causality-driven reactive coupling enabling non-Hermitian pumping of the dark sector. We derive the optimal condition for this process and show that it corresponds to the supercritical coupling regime previously identified in [Nature 626, 765 (2024)], while naturally recovering universal quasi-BIC asymmetry scaling. Extending the theory to Dirac-like dispersions in photonic crystal slabs, we identify an open-Dirac singularity where the Dirac gap matches the supercritical regime. A four-wave Hamiltonian quantitatively reproduces rigorous coupled-wave analysis, revealing the breakdown of conventional critical coupling. Near this regime, absorptive cross-coupling induces coherent absorption interference and enables suppression of effective dissipative losses beyond conventional material limits. These results motivate the concept of a generalized bound state in the continuum (gBIC) as a limiting non-Hermitian state where radiative and effective gain compensate, producing a true divergence of the total quality factor. Overall, this work establishes a unified framework connecting BIC interference, Dirac topology, and non-Hermitian physics for ultra-high-Q enhancement and loss engineering in open photonic systems.