## 2 Theoretical Approach: [PERSON]'s Function  
We consider a 1D quasi-periodic structure composed of two blocks \(A\) and \(B\) stacked following the Fibonacci sequence [29]: \(S_{k+1}=S_{k}S_{k-1}\) with the initial conditions \(S_{1}=A\) and \(S_{2}=AB\), where \(S_{k}\) is the \(k\)th generation of the Fibonacci sequence. For exemple \(S_{3}=ABA\), \(S_{4}=ABAAB\), \(S_{5}=ABAABABA\),... For the Fibonnaci structures considered here (Figure 1), the block \(A\) is formed of identical horizontal and vertical coaxial cables of length \(d_{2}=d_{1}\) (Figure 1a), while the Block \(B\) is similar to the block \(A\) but with different length of its vertical cable, namely \(d_{3}=d_{2}/2\) (Figure 1b). All coaxial cables are characterized by the same impedance \(Z\). The localization and the propagation of electromagnetic waves in the Fibonacci structures depicted in Figure 1f,g, are performed within the framework of the interface response theory of continuous media [75]. This theory enables to calculate the Green's function of any heterogenous system containing a large number of interfaces separating different homogenous media. In what follows, the fundamental equations and the basic concepts of this theory [75] are presented. We consider an heterogenous system composed of \(N\) different homogeneous sub-systems defined in their domains \(D_{i}\) (\(D=\bigcup D_{i}\) being the whole space of the system). Each subsystem \"\(i\)\" is connected in general to \(j\) (\(1\leq j\leq J\)) other pieces through sub-interface domains \(M_{ij}\) (\(M_{i}=\bigcup M_{ij}\) being the interface space of the sub-system \"\(i\)\"). All these interface spaces \(M_{i}\) constitutes what is called the interface space \(M\) of the heterogenous material. In this theory, the elements of the Green's function \(g(DD)\) of a composite material is given by [75]  
\[g(DD) = G(DD)-G(DM)G^{-1}(MM)G(MD)\] \[+G(DM)G^{-1}(MM)g(MM)G^{-1}(MM)G(MD),\]  
Figure 1: Schematic representation of blocks \(A\) and \(B\) constituting the Fibonacci structure. (**a**) The block \(A\) consists of a horizontal coaxial cable of length \(d_{1}\) combined with another identical vertical stub of length (\(d_{2}=d_{1}\)). (**b**) The Block \(B\) is similar to the block \(A\) but with the length of the vertical stub of length \(d_{3}=d_{2}/2\). All cables are characterized by the same impedance \(Z\). (**c–e**) give the shape of the Fibonacci sequences \(S_{3}\), \(S_{4}\) and \(S_{5}\) respectively. (**f**) A finite Fibonacci structure (6 th generation) inserted horizontally between two semi-infinite waveguides. (**g**) A finite Fibonacci structure (5 th generation) grafted vertically along a waveguide. where \(G(DD)\) is the reference Green's function constituted from the superposition of the truncated bulk Green's functions of each piece defined in its space domain \(D_{i}\). \(g(MM)\) corresponds to the Green's function of the composite system in the whole interface space \(M\). The inverse of \(g(MM)\) enables to calculate the dispersion relations of the composite system using the following expression [75]  
\[det[g^{-1}(MM)]=0. \tag{2}\]  
Also, using this theory, Equation (1) enables to calculate the eigenvectors \(u(D)\) of the composite material from the relation [75]  
\[u(D) = U(D)-U(M)G^{-1}(MM)G(MD)+\] \[U(M)G^{-1}(MM)g(MM)G^{-1}(MM)G(MD),\]  
where \(U(D)\), \(U(M)\) and \(u(D)\) are row vectors. In particular, if \(U(D)\) represents a bulk wave launched in one homogeneous piece of the whole system [67], then Equation (3) enables to calculate the transmission and reflection coefficients through the system. The calculation of the Green's function of a FS in the whole interface space \(M\), requires the knowledge of the surface elements of its elementary constituents, namely, the Green's function of the block \(A\) (Figure 1a) and of the block \(B\) (Figure 1b). The explicit expressions of the \(2\times 2\) matrices of these two blocks are given by Equations (A6) and (A7) in the Appendix A.