## 4 Numerical and Experimental Results
### Vertical Fibonacci Sequence: Confined and Surface Modes  
In the following, we will discuss the case where the Fibonacci sequences are grafted vertically to an horizontal waveguide (Figure 1g). The choice of this geometry is motivated by the fact that the maxima and minima of the transmission spectra permit to deduce all the eigenmodes of the structure [70,71]. As mentioned in Section 2, the transmission maxima occur for the eigenfrequencies of the finite FS when the boundary conditions at its both ends are perfect magnetic conductor type (i.e., vanishing magnetic field). However, the transmission minima occurs for eigenfrequencies of the finite Fibonacci sequence when the boundary conditions are perfect magnetic conductor type on the free surface of the structure and perfect electric conductor type (i.e., vanishing electric field) on the other side which will be grafted on the  
frequency \(f_{c}=53\) MHz is displayed in Figure 9b-d. Indeed, a magnification of Figure 7 around \(f_{c}\) (Figure 9a) shows stable modes. The behavior of the electric field of such modes are not similar to surface modes (Figure 8), but exhibit rather a self similar behavior as for the horizontal Fibonacci structure (Figure 4). This is illustrated in Figure 9b-d for the mode labeled 6 (12 th generation) as compared to the modes labeled 7 and 8 for the 11 th and 10 th generations respectively. However, one can notice that the self-similarity occurs here every generation instead of three generations as it is the case in Figure 4, where the range on the horizontal axis is adapted for comparison to 11 th and 10 th FS generations. Figure 6: Theoretical (solid curves) and experimental (open circles) variations of the transmittance versus the frequency for a Fibonacci sequence inserted vertically along a waveguide (Figure 1g) for the generations \(S_{3}\)–\(S_{9}\). Figure 7: Dispersion of the eigenfrequencies (dots) of different generation numbers. The boundary conditions on both sides of the Fibonacci sequence (FS) are H = 0. One can notice the existence of primary and secondary gaps. All the discrete modes are obtained from the maxima of the transmission spectra displayed in Figure 6. The labels 1, 2, 3, 4 and 5 represent the surface modes. Figure 8: (**a–c**) Spatial representation of the LDOS (in arbitrary units) of the modes labeled 1, 2, 3 in Figure 7 at \(f_{1}=26.85\) MHz (**a**), \(f_{2}=62.10\) MHz (**b**) and \(f_{3}=51.38\) MHz (**c**) respectively. Dashed (solid) lines correspond to the 6 th (7 th) generation. (**d**,**e**) Same as (**a–c**) but for the modes labeled 4 and 5 in Figure 7 at \(f_{4}=45.56\) MHz (**d**) and \(f_{5}=71.93\) MHz (**e**), respectively. Dashed (solid) lines correspond to the 7 th (9 th) and 8 th (10 th) generation, respectively. As mentioned in Section 2, the minima of the transmittance plotted in Figure 6 give the eigenmodes of the finite FS with vanishing electric field on one side and magnetic field on the other side. These minima are reported in Figure 10 for different generations. One can notice that the eigenmodes of this system are quite different from those of the FS with vanishing magnetic field on both sides (Figure 10). Moreover, the primary and secondary gaps appear as the generation number increases and some of these gaps exhibit surface modes denoted \(1^{\prime}\) and \(2^{\prime}\) for all generations. These modes are induced by the short-circuited surface as illustrated by the spatial localization of the electric field displayed in Figure 11 for these two modes. One can see clearly a decrease of the LDOS (or the square modulus of the electric field) of these modes far from the surface when penetrating inside the bulk of the Fibonacci system with almost the same localization length (solid and dashed curves).