  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.
