Photonic band gap structures in the Thue-Morse lattice

Abstract

In this paper, we investigate the physical mechanism and properties of the two classes of photonic gaps in the aperiodic Thue-Morse (TM) lattice: the traditional gaps and the fractal gaps. Based on the analysis of the field distributions of the midgap and gap-edge states, we demonstrate that the two classes of gaps originate from the different spatial correlations. For the case of the traditional gaps the correlation mechanism is confirmed by the approximated transfer matrix method, in which only the relevant interfaces are considered. Then we find that the edges of the traditional gaps can be obtained from the condition ${x}_{2}=2$, where ${x}_{2}$ is the second order trace map. Around the basic traditional gap, there are two finite spectral regions in which the transmission spectra look like the bands of a periodic system and the eigenstates behave like the Bloch states, so that they are defined as the traditional bands. Simultaneously, it is numerically illuminated that the traditional bands are well defined physically. Finally, the localization properties of the fractal gap-edge state are studied by the quality factor $Q$. The result indicates the fractal gap-edge states become hyperexponentially localized as the system size increases.