## Appendix A Inverse Surface Green's Function of the Fibonacci Sequence  
To build the Green's function of any heterostructure based on finite coaxial cables, we need first the Green's function of an infinite homogeneous isotropic dielectric cable characterized by its characteristic impedance \(Z_{i}\) and its relative permittivity \(\varepsilon_{i}\). The Green's function between two points \(x\) et \(x^{\prime}\) of the infinite cable is given by [75]  
\[G_{i}(x,x^{\prime})=-\frac{e^{-\alpha_{i}|x-x^{\prime}|}}{2F_{i}}, \tag{1}\]  
with  
\[\alpha_{i}=-j\frac{\omega}{c}\sqrt{\varepsilon_{i}}\;\;\text{and}\;\;F_{i}=-j \frac{\omega}{Z_{i}}. \tag{2}\]  
\(c\) the speed of light in vacuum, \(\omega\) is the angular frequency of the wave and \(j=\sqrt{-1}\). Before treating the problem of the quasi-periodic structure, it is necessary to establish the Green's function of its elementary constituents in their own space of interface, namely, the surface elements of a finite cable of length \(d_{1}\), the stubs 2 and 3 of lengths \(d_{2}\) and \(d_{3}\) respectively and the semi-infinite wire \(s\). The finite cable is limited by two free surfaces (with H = 0 boundary condition) defined at \(x=-\frac{d_{1}}{2}\) and \(x=+\frac{d_{1}}{2}\). The inverse Green's function \([g_{1}(MM)]^{-1}\) of this system can be written in the form of a \((2\times 2)\) matrix within the interface space \(M_{i}=\{-\frac{d_{1}}{2},+\frac{d_{1}}{2}\}\) as follows [67]  
\[[g_{1}(MM)]^{-1}=\left( \begin{array}{cc}-\frac{F_{i}C_{1}}{S_{1}}&\frac{F_{ i}}{S_{1}}\\ \frac{F_{1}}{S_{1}}&-\frac{F_{1}C_{1}}{S_{1}}\end{array} \right). \tag{3}\]  
When the boundary condition on the upper end of the stubs 2 and 3 (Figure 1) is of type \(E=0\), the inverse of the Green's function on the lower end is given by [67]  
\[g_{i}(-d_{i}/2,-d_{i}/2)^{-1}=-\frac{F_{i}C_{i}}{S_{i}} \tag{4}\]  
with \(C_{i}=\cosh(\alpha_{i}d_{i})\) and \(S_{i}=\sinh(\alpha_{i}d_{i})\) (i = 1, 2, 3) (see Figure 1). The inverse of the surface element of the semi-infinite cables \(s\) surrounding the FS structure, is given by [67]  
\[g_{s}(0,0)^{-1}=-F_{s} \tag{5}\]  
where \(F_{s}=-j\omega/Z_{s}\) and \(Z_{s}\) is the characteristic impedance of the semi-infinite cables. From Equations (3) and (4), one can deduce easily the inverse Green's functions \([g_{A}(MM)]^{-1}\) and \([g_{B}(MM)]^{-1}\) of blocks \(A\) and \(B\), respectively as follows  
\[[g_{A}(MM)]^{-1}=\left( \begin{array}{cc}a_{A}&b_{A}\\ b_{A}&c_{A}\end{array} \right) \tag{6}\]and  
\[[g_{B}(MM)]^{-1}=\left( \begin{array}{cc}a_{B}&b_{B}\\ b_{B}&c_{B}\end{array} \right), \tag{10}\]  
with \(a_{A}=a_{B}=-\frac{F_{1}C_{1}}{S_{1}}\), \(b_{A}=b_{B}=\frac{F_{1}}{S_{1}}\), \(c_{A}=-\frac{F_{1}C_{1}}{S_{1}}-\frac{F_{2}C_{2}}{S_{2}}\) and \(c_{B}=-\frac{F_{1}C_{1}}{S_{1}}-\frac{F_{2}C_{3}}{S_{3}}\). The inverse Green's function of a given Fibonacci sequence can be built as a superposition of those of blocks \(A\) and \(B\) arranged following the Fibonacci rule. The space of interface \(M\) is composed of all the connection points between the different finite size cables constituting the system. The inverse of the matrix \(g(MM)^{-1}\) giving all the interface elements of the Green's function \(g\) is a finite tridiagonal matrix formed by a linear superposition of the above elements \(g_{(i=A,B)}(M_{i}M_{i})^{-1}\) (Equations (10) and (10)). For example, in the case of the fourth Fibonacci structure \(S_{4}=ABAAB\), the matrix \(g(MM)^{-1}\) can be written in the following tri-diagonal matrix form:  
\[[g_{FS}(MM)]^{-1}=\left( \begin{array}{cccccc}a_{A}&b_{A}&0&0&0&0\\ b_{A}&c_{A}+a_{B}&b_{B}&0&0&0\\ 0&b_{B}&c_{B}+a_{A}&b_{A}&0&0\\ 0&0&b_{A}&c_{A}+a_{A}&b_{A}&0\\ 0&0&0&b_{A}&c_{A}+a_{B}&b_{B}\\ 0&0&0&0&b_{B}&c_{B}\end{array} \right).