Plasmonic Fourier Surfaces Revisited: Relating Bandgaps with Bound States in the Continuum
Periodically corrugated metal interfaces supporting surface plasmon polaritons (SPPs) belong to the earliest nanoplasmonic platforms. Even simplest reliefs described by a few harmonics - plasmonic Fourier surfaces - display markedly different far-field signatures depending on the corrugation depth and symmetry: shallow reliefs exhibit a plasmonic bandgap (PBG) between two hybridized SPP standing waves, while deeper reliefs support asymmetry-induced single sharp resonances, termed over the last decade as quasi-bound states in the continuum (qBICs). Although these spectral features have long been observed experimentally and treated empirically, the underlying eigenmode evolution connecting the shallow and deep corrugation regimes has remained largely unexplored. Here we revisit this long-standing problem by analyzing it in terms of modern eigenstate formalism. Starting from the Rayleigh hypothesis, we develop a concise first-principle analytical description that explicitly captures how one eigenmode transforms from a dark bound state into an observable qBIC, while the other turns from bright into an overcoupled, unobservable state - thus unifying the SPP manifestations featuring PBG and qBIC within the same eigenmode framework. Finally, we demonstrate the practical relevance of the theory by showing how precise eigenstate engineering can enhance the SPP refractive index sensitivity.