Resonance and topological singularity near and beyond zero frequency for waves: model, theory, and effects

Research interest in resonance and topology for systems at near-zero frequency, whose wavelength could be 2 orders larger than the scale of resonators is very rare, since the trivial effective-medium theory is generally thought to be correct in this regime. Also, the complex frequency regime is generally thought to be irrelevant to the topological properties of Hermitian systems. In this work, we find the general conditions to realize near-zero frequency resonance for a resonator and theoretically propose two kinds of realizations of such resonators, which are confirmed by numerical methods. The photonic crystals with such a resonator as the unit cell present rich topological characteristics at the near-zero frequency regime. The topological singularity that corresponds to the resonant frequency of the unit cell can be pushed to zero frequency at the bottom of the first band by tuning a certain parameter to a critical value. Surprisingly, we find that, when the parameter is tuned over the critical value, the singularity has disappeared in the first band and is pushed into the imaginary frequency regime, but now the topology of the first band and gap is still nontrivial, which is demonstrated by the existence of the topological edge state in the first gap, the negative sign of imaginary part of the surface impedance, and the symmetry property of Wannier functions. So, we are forced to accept that the singularity in the imaginary frequency regime can influence the topology in the real frequency regime. So, for the first time, to the best of our knowledge, we find that the singularity in the pure imaginary regime can still cause the observable topological effects on the real frequency regime, even for the Hermitian systems. Now, zero frequency acts as a novel exceptional point for Hermitian systems and the topology of the first band and first gap could be quite different from other bands and gaps, since they are intrinsically connected with zero frequency. Other new phenomena are also observed when the singularity is at the near-zero frequency regimes (real or imaginary), e.g.,  the cubic relationship between reflection coefficient and the frequency, the robust wide-bandwidth high transmission at very low frequency, etc. Besides the theoretical importance, some basic applications, such as the robust deep subwavelength wide bandwidth high-transmission layered structures, the subwavelength wide bandwidth absorbers, and the cavity from the topological subwavelength edge state are proposed, which can inspire new designs in many areas of optics, microwaves, and acoustics. This work opens a new window for rich topological physics and revolutionary device designs at the near and beyond zero-frequency regimes.