3.1 Generalized Thue-Morse sequence

A one dimensional GTM sequence is called aperiodic because it is more disordered than the quasiperiodic one. In addition, the two different materials included in one dimensional GTM system should be structured by applying the substitution rule: <sup>σ</sup>GTMðH; <sup>L</sup><sup>Þ</sup> : <sup>H</sup> ! <sup>H</sup>mLn;<sup>L</sup> ! <sup>L</sup>mHn [16], where <sup>H</sup> and <sup>L</sup> represent the two layers, having the higher and the lower refractive indices, respectively. We note that the Fourier spectra of the GTM sequence is singular and continuous. Also, the GTM quasiperiodic chain is generated by <sup>a</sup> recursive deterministic sequence Sk+1 <sup>n</sup> <sup>n</sup> verifying: Skþ<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>m</sup> <sup>k</sup> Sk , where Sk is the conjugated sequence of S<sup>n</sup> k, m and n are the parameters of GTM sequence with order k. This rule can be applied to two dimensions: horizontally and vertically.

Based on GTM sequence Skþ1, we give Table 1 which illustrates an example of organized multilayered stacks (H, L) for m = n = 2.

The configuration of the proposed 1D photonic dielectric/quasiperiodic superconducting layers which is built according to the GTM sequence is shown in Figure 1.
