  We demonstrate a photonic analog of twisted bilayer graphene that has
ultra-flat photonic bands and exhibits extreme slow light behavior. Our twisted
bilayer photonic device, which has an operating wavelength in C-band of the
telecom window, uses two crystalline silicon photonic crystal slabs separated
by a methyl methacrylate tunneling layer. We numerically determine the magic
angle using a finite-element method and the corresponding photonic band
structure, which exhibits a flat band over the entire Brillouin zone. This flat
band causes the group velocity to approach zero and introduces light
localization in a linear periodic photonic system. Using a plane-wave continuum
model we find that photonic modes in twisted bilayer photonic crystals are not
as tightly bound as their electronic counterparts in twisted bilayer graphene.
In addition, the photonic system has a larger band asymmetry. The band
structure can easily be engineered by adjusting the device geometry, giving
significant freedom in the design of devices. Our work provides a fundamental
understanding of the photonic properties of twisted bilayer photonic crystals
and opens the door to nanoscale-based non-linear enhancement and superradiance.
