The evolution of topological singularities between real- and complex-frequency domains and the engineering of photonic bands for Hermitian and non-Hermitian photonic crystals

Abstract Singularity annihilation, generation, and evolving (SAGE) lead to the topological phase transition (TPT) in electronic, photonic acoustic systems. Traditionally singularity study of Hermitian systems is only focused on real frequency domain. In this work, we systematically investigate complicated SAGE complex domain (CFD) for one-dimensional (1D) non-Hermitian a more general picture revealed. First, abnormal phenomenon that one evolves from first band zero then into pure imaginary 1D crystals (PhCs). New results, e.g. condition at frequency, stricter definition Zak singularities are pushed found. Second, CFD constructed. Complicated processes observed, such as not axis but also axis, closed loops which connected imaginary-frequency real-frequency axis. Even more, when PhCs degenerated since permittivity kind layer becomes same neighbor layer, integral reduced will move infinite far away come back with half-integral shift. Third, gain or absorption introduced in, tilted observed. The moving means stable states resonance could be realized. Such opens new window studies TPT rich phenomena expected. Besides theoretical importance, evolution can used engineer properties PhCs. Some novel applications, super-broadband sub-wavelength high-transmission layered structure broadband deep-sub-wavelength absorber, proposed.