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

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