## 2 Theoretical Approach: [PERSON]'s Function
Also, the method for deriving the inverse surface Green's function matrix of the whole FS (Figure 1c-e) is described in the Appendix A for the 4 th generation for example. \(g_{FS}^{-1}(M_{e}M_{e})\) can be written as  
\[[g_{FS}(M_{e}M_{e})]^{-1}=\left( \begin{array}{cc}a&b\\ b&c\end{array} \right), \tag{4}\]  
where \(M_{e}=\{0,L\}\) is the space of interfaces at both free extremities of a given FS and \(L\) is the total length of the structure. \(a\), \(b\) and \(c\) are real quantities, functions of the different parameters of the constituent's elements. The derivation of these parameters is described in Equations (A6) and (A7) of the Appendix A. Let us consider the finite FS structure delimited by two homogeneous semi-infinite waveguides labeled \(s\) characterized by their inverse surface Green's function at the surface \(x=0\) or \(x=L\) (see Equation (A5) in the Appendix A)  
\[g_{s}(0,0)^{-1}=g_{s}(L,L)^{-1}=-F_{s}, \tag{5}\]  
where \(F_{s}=-j\omega/Z_{s}\), \(Z_{s}\) is the impedance of semi-infinite waveguide and \(\omega\) is the angular frequency. Then the inverse Green's function of the whole horizontal composite system (Figure 1f) can be obtained from a superposition of the matrix given in Equation (4) surrounded by two semi-infinite waveguides on both sides (Equation (5)) such as  
\[[g_{h}(M_{e}M_{e})]^{-1}=\left( \begin{array}{cc}a-F_{s}&b\\ b&c-F_{s}\end{array} \right). \tag{6}\]  
Similarly, the inverse Green's function of the whole vertical system (Figure 1g) can be obtained from a superposition of the matrix given in Equation (4) surrounded by two semi-infinite waveguides on one side (Equation (5)) such as  
\[[g_{v}(M_{e}M_{e})]^{-1}=\left( \begin{array}{cc}a-2F_{s}&b\\ b&c\end{array} \right). \tag{7}\]Here \(h\) and \(v\) stand for horizontal and vertical insertion of the finite FS. As a preliminary remark, let us notice from Equations (2) and (7) that one can deduce the expressions giving the eigenmodes of the Fibonacci sequence in two particular cases:  
1. when the magnetic field vanishes on both sides of the FS structure (i.e., \(Z_{s}\rightarrow\infty\) or equivalently \(F_{s}=0\)), one obtains: \[ac-b^{2}=0.\] (8)
2. When the electric field vanishes on the side of the structure which will be grafted on the waveguide and the magnetic field vanishes on the free surface (i.e., \(Z_{s}=0\) or \(F_{s}\rightarrow\infty\)), one obtains: \[c=0.\] (9)  
Now, the determination of the properties of wave propagation and localization in a Fibonacci system inserted (grafted) horizontally (vertically) between two homogeneous waveguides labeled \(s\) (Figure 1f,g), requires the calculation of the transmission coefficient of an incident plane wave launched from the left semi-infinite waveguide and traveling through the system. From the bulk Green's function (Equation (A1)) and Equations (6) and (7), one can get the transmission functions for the horizontal and vertical structures such as: \(t_{h}=-2F_{s}g_{h}(0,L)\) and \(t_{v}=-2F_{s}g_{v}(0,0)\) respectively. Therefore, one obtains  
\[t_{h}=\frac{-2F_{s}b}{ac-b^{2}-(F_{s})^{2}-(a+c)F_{s}}\, \tag{10}\]  
and  
\[t_{v}=\frac{-2F_{s}c}{ac-b^{2}-2F_{s}c}. \tag{11}\]  
Let us notice that from Equation (11), the eigenmodes of the FS with the boundary conditions of type \(E=0\) on one side and \(H=0\) on the other side (Equation (9)) are obtained from \(t_{v}=0\) (i.e., \(c=0\)) and the eigenmodes of the FS with the boundary conditions of type \(H=0\) on both sides (Equation (8)) are obtained from \(t_{v}=1\) (i.e., \(ac-b^{2}=0\)).