Photonic Floquet media with a complex time-periodic permittivity

We study the exceptional point (EP) phenomena in a photonic medium with a complex time-periodic permiitivity, i.e., $\epsilon(t)=\epsilon_o+\epsilon_r*sin(\Omega t+\phi)$. We formulate the Maxwell's equations in a form of first-order non-Hermitian Floquet Hamiltonian matrix and solve it analytically for the Floquet band structures. In the case when $\epsilon_r$ is real, to the first order in $\epsilon_r$, the band structures show a phase transition from an exact phase with real quasienergies to a broken phase with complex quasienergies inside a region of wave vector space, the so-called k-gap. We show that the two EPs at the upper and lower edges of the k-gap have opposite chiralities in the stroboscopic sense. Thus, by picking up the mode with a positive imaginary quasienergy, the wave propagation inside the k-gap can grow exponentially. In three dimensions, such pairs of EPs span two concentric spherical surfaces in the $\vec{k}$ space and repeat themselves periodically in the quasienergy space with Omega as the period. However, in the case when $\epsilon_r$ is pure imaginary, the k-gap disappears and gaps in the quasienergy space are opened. Our analytical results agree well with the finite difference time domain (FDTD) simulations. To the second order in $\epsilon_r$, additional EP pairs are found for both the cases of real and imaginary $\epsilon_r$.