Complex band structure of one-dimensional polariton crystal

The exceptional point (EP), at which the relevant eigenvalues and eigenstates are simultaneously identical, typically exists in non-Hermitian systems with parity-time (PT) symmetric complex potentials, and gives rise to many intriguing behaviors in various physical realms. In this work, we explore the complex band structure of one-dimensional "polariton crystals" that can be constructed in waveguide-resonator coupled systems, with PT-symmetric potential. Analysis based on the transfer matrix and the coupled mode theory shows that the complex band structure is intimately determined by the interaction between the Bragg resonance and the polariton one, the gain/loss coefficients, in addition to the coupling strength. A miniband is induced due to the interaction of these two resonances, which is a defect-like band and appears quite different for the band structure evolution. Furthermore, PT-symmetric phase transition occurs in the momentum space for certain amounts of non-Hermiticity. As the non-Hermiticity increases, the EP formed in the original polariton gap approaches another EP formed at the touch point of the folded Bragg bands (where the thresholdless transition occurs). Then they coalesce at a specific non-Hermiticity, and finally disappear. Subsequently, the transmission spectra of such polariton crystals show intriguing phenomena induced by the EPs. Our results provide a different perspective to understand PT-symmetric polariton crystals and may find applications in gain/loss induced lasing by 'polaritons'.