One-dimensional time-Floquet photonic crystal

Using the Floquet Hamiltonian derived based on the time-dependent perturbation theory, we investigated the quasienergy bands of a one-dimensional time-Floquet photonic crystal. The time-Floquet photonic crystal contains two alternating layers labeled as A and B, and the permittivity of A layer is modulated periodically in time. We showed that the quasienergy bands are reciprocal when the modulation function is a function of time only, while the quasienergy bands could be nonreciprocal when the permittivity is modulated in both time and space through an unique combination. In the former case, the coupling between the positive (negative) and positive (negative) bands results in quasienergy gaps, while the coupling between the positive and negative bands leads to pairs of exception points, when the modulation is on the real part of the permittivity. In the latter case, the coupling between the positive (negative) and positive (negative) bands still results in quasienergy gaps. However, the coupling between the positive and negative bands leads to quasienergy gaps at a small modulation speed and pairs of exceptional points at a high modulation speed.