Tuning band structures of photonic multilayers with positive and negative refractive index materials according to generalized Fibonacci and Thue-Morse sequences.

We investigate the photonic band structure, using the transfer-matrix method, in one-dimensional structures composed of a dispersive metamaterial A juxtaposed with a non-dispersive dielectric B according to the generalized Fibonacci and Thue-Morse sequences, which are ruled and characterized by two positive integer numbers, p  and q. We present the band structures of different generations of these sequences for both TE and TM modes and discuss how they are affected by the p  and q parameters and the ratio between the thicknesses of the building blocks. We obtain a new and very important analytical expression for the frequency in which the average refraction index vanishes, i.e. when the [Formula: see text] gap condition is satisfied, and we investigate, in detail, the behavior of this emergent scale insensitive gap, as a function of the thicknesses ratio, for all structures considered. For the metamaterial considered, we show that the reduced frequency for [Formula: see text] converges faster or slower to 0.41 depending on the sequence. By adjusting the p  and q parameters, we can obtain a higher concentration of the pass bands inside (outside) the n   <  0 region when there are more (less) blocks A than B in the unit cell. Our results also show the presence of omnidirectional and complete band gaps, which have various practical and technological applications.