Band rules for the frequency spectra of three kinds of aperiodic photonic crystals with negative refractive index materials

By means of the theory of electromagnetic wave propagation and transfer matrix method, this paper investigates the band rules for the frequency spectra of three kinds of one-dimensional (1D) aperiodic photonic crystals (PCs), generalized Fibonacci GF(p,1), GF(1,2), and Thue--Morse (TM) PCs, with negative refractive index (NRI) materials. It is found that all of these PCs can open a broad zero-\barn gap, TM PC possesses the largest zero-\barn gap, and with the increase of p, the width of the zero-\barn gap for GF(p,1) PC becomes smaller. This characteristic is caused by the symmetry of the system and the open position of the zero-\barn gap. It is found that for GF(p,1) PCs, the possible limit zero-\barn gaps open at lower frequencies with the increase of p, but for GF(1,2) and TM PCs, their limit zero-\barn gaps open at the same frequency. Additionally, for the three bottom-bands, we find the interesting perfect self-similarities of the evolution structures with the increase of generation, and obtain the corresponding subband-number formulae. Based on 11 types of evolving manners Q_i (i = 1, 2, ...., 11) one can plot out the detailed evolution structures of the three kinds of aperiodic PCs for any generation.