In this work, we study the photonic band of cumulative Fibonacci lattices, of which the structure is composed of all generated units in a Fibonacci sequence. The results are compared with distributed Bragg reflector (DBR) structures with the same numbers of layers. Photonic bandgaps are found at two characteristic frequencies, symmetrically separated from the central bandgap in the DBR counterpart. Field amplitude and phase distribution in the Fibonacci lattice indicates an interferential origin of the bandgaps. Fourier transform on the refractive index profile is carried out, and the result confirms a determinate long-range periodicity that agrees well with the photonic band structure.Province of China (Q2015002).