Designing a Transition Photonic Band with a Synthetic Moire Sphere

In recent years, twisted bilayer graphene has become a hot topic and inspired the research upsurge of photonic moir\'e lattice. Here, we designed a photonic moir\'e superlattice with two synthetic twist angles and constructed a synthetic moir\'e sphere based on these two angles. Thus, we have more degrees of freedom to design the band structure flexibly. A type of transition photonic bands (TPBs) is obtained in such a moir\'e superlattice. We investigate the influence of two twist angles on TPBs and find a series of magic angle pairs with optimal band compression of TPB. The interesting optical properties of TPBs are experimentally demonstrated, including pulse delay, nonlinear optical enhancement, and pulse width compression. Our work introduces a new path to explore multi-twist angles moir\'e superlattices and reveals that the designed photonic moir\'e superlattice based on moir\'e spheres has broad application prospects including optical signal processing, nonlinear optics processes ,and other light-matter interactions.