  Periodically driven systems, characterised by their inherent non-equilibrium
dynamics, are ubiquitously found in both classical and quantum regimes. In the
field of photonics, these Floquet systems have begun to provide insight into
how time periodicity can extend the concept of spatially periodic photonic
crystals and metamaterials to the time domain. However, despite the necessity
arising from the presence of non-reciprocal coupling between states in a
photonic Floquet medium, a unified non-Hermitian band structure description
remains elusive. Here, we experimentally reveal the unique Bloch-Floquet and
non-Bloch band structures of a photonic Floquet medium emulated in the
microwave regime with a one-dimensional array of time-periodically driven
resonators. Specifically, these non-Hermitian band structures are shown to be
two measurable distinct subsets of complex eigenfrequency surfaces of the
photonic Floquet medium defined in complex momentum space. In the Bloch-Floquet
band structure, the driving-induced non-reciprocal coupling between oppositely
signed frequency states leads to opening of momentum gaps along the real
momentum axis, at the edges of which exceptional phase transitions occur. More
interestingly, we show that the non-Bloch band structure defined in the complex
Brillouin zone supplements the information on the morphology of complex
eigenfrequency surfaces of the photonic Floquet medium. Our work paves the way
for a comprehensive understanding of photonic Floquet media in complex
energy-momentum space and could provide general guidelines for the study of
non-equilibrium photonic phases of matter.
