  Drawing inspiration from bilayer graphene, this paper introduces its photonic
analog comprising two stacked graphene-like photonic crystals, that are coupled
in the near-field through spoof surface plasmons. Beyond the twist degree of
freedom that can radically alter the band structure of the bilayer photonic
graphene, the photonic dispersion can be also tailored via the interlayer
coupling which exhibits an exponential dependence on the distance between the
two photonic crystals. We theoretically, numerically, and experimentally
characterize the band structures of AA- and AB-stacked bilayer photonic
graphene, as well as for twisted bilayer photonic graphene with even and odd
sublattice exchange symmetries. Furthermore, we numerically predict the
existence of magic angles in bilayer photonic graphene, which are associated
with ultra-flat bands resulted from interlayer hybridization. Finally, we
demonstrate that the bilayer photonic graphene at a particular twist angle
satisfying even sublattice exchange symmetry is a high-order photonic
topological insulator. The proposed bilayer photonic graphene could constitute
a useful platform for identifying new quantum materials and inspiring
next-generation photonic devices with new degrees of freedom and emerging
functionality.
