Flatbands and Localization in Moir\'e Superlattice of Bilayer Photonic crystal.

We investigate the physics of the photonic band structures of the moir\'e patterns that emerged when overlapping two mismatched period uni-dimensional (1D) photonic crystal slabs. The band structure of our system is a result of the interplay between intra-layer and inter-layer coupling mechanisms, which can be fine-tuned via the distance separating the two layers. We derive an effective Hamiltonian that captures the essential physics of the system and reproduces all numerical simulations of electromagnetic solutions with high accuracy. Most interestingly, \textit{magic distances} corresponding to the emergence of photonic flat bands of the moir\'e lattice are observed. We demonstrate that these flat band modes at a magic distance are tightly localized within a moir\'e period. Moreover, we suggest a single-band tight-binding model that describes the moir\'e minibands, of which the tunnelling rate can be continuously tuned via the inter-layer strength. Our results show that the band structure of bilayer photonic moir\'e can be engineered in the same fashion as the electronic/excitonic counterparts. It would pave the way to study many-body physics at photonic moir\'e flat bands and novel optoelectronic devices.