Abstract
We theoretically demonstrate how to acquire the full bandgap spectrum of a photonic lattice of arbitrary profile. Focusing on the 1D case in the presence of a linear refractive index acceleration and employing a multiscale analysis, we show that each photonic band can be directly mapped by the light evolution in the lattice, whereas the size of each gap corresponds to the tunneling rate. We verify the analytical results with numerical simulations and discuss experimental realizations of this technique for dispersive spectroscopy.