Recently, it has been revealed that a non-Hermitian optical system having gain/loss shows very unique characteristics around so-called "exceptional points" when it has a certain symmetry of complex refractive index, such as parity-time(PT)symmetry. In this article, we describe band dispersion and topological properties of non-Hermitian optical periodic systems(non-Hermitian photonic crystals). We show that they exhibit exotic dispersion around exceptional points including superluminal states, and also prove that it is possible to realize nontrivial topology solely by gain/loss modulation of a simple periodic array. In addition, it is revealed that non-Hermitian systems exhibit a much richer variety of topological properties governed by their complex bandgaps and multiple topological invariants, in contrast to Hermitian counterparts. Practically, this means that one can control dispersion and topology by modulating the imaginary part of refractive index via current injection. This novel controllability paves the way to reconfigurable topological photonics, which may lead to novel optical processing circuits.
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