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In this article, we developed a generalized coupled-mode theory for mixing anisolated state with the gapped continuum to unify the engineering mechanism ofgap-protected zero modes, Fano resonance and Tamm surface states, even thoughthose phenomena are diverse in numerous branches of physics and optics. Bytuning the on-site potential and coupling strength of/with the isolated state,we found the operating characteristics from zero modes, Fano resonance and toTamm modes, with explicitly demonstrating their localization, transmissionspectra and distinct evolution dynamics of the 'bound states in the gappedcontinuum' (BIGC). As an extent, we designed two sandwich-like structures inthe BIGC framework: (1) the 'isolated-continuous-isolated' (ICI), which leadsto adiabatic elimination technique and topologically-protected edge states, and(2) the 'continuous-isolated-continuous' (CIC), which results into the domainwall dynamics. Finally, we proposed the optical waveguide array (WA) modellingof the modified Su-Schrieffer-Heeger dimerized lattice to achieve thoseinteresting engineering processes, and meanwhile to shed light on thegeneralization of bound states in the continuum into photonics and condensedmatters.

Engineering zero modes, Fano resonance and Tamm surface states of 'bound states in the gapped continuum'