## 4 Numerical and Experimental Results
### Horizontal Fibonacci Sequence: Scaling Law  
In this subsection, we will discuss the properties of wave propagation and localization in a Fibonacci sequence sandwiched horizontally between two semi-infinite waveguides (Figure 1f). To our knowledge, neither theoretical nor experimental studies have been performed on photonic crystals using such quasi-periodic systems. Figure 3b-g give the transmission coefficient versus the frequency for the generations \(S_{4}\), \(S_{5}\), \(S_{6}\), \(S_{7}\), \(S_{8}\) and \(S_{9}\) composed of 5, 8, 13, 21, 34 and 55 blocks, respectively. Solid, dashed and open circles lines have the same meaning as defined in Figure 2. In this figure, one can distinguish two regions of frequencies: the regions where the transmission presents clear dips as the generation number increases, which correspond to the forbidden bands, and the regions where the transmission is more pronounced, which corresponds to the allowed bands. From these results, it can be derived that the transmission for the Fibonacci structures (Figure 3b-g) has more than one gap (transmission dip) with respect to the transmission spectrum of the periodic system composed by an alternate repetition of \(A\) and \(B\) blocks (Figure 3a), especially for increasing generations of FS (Figure 3c-g). This phenomenon is the result of the fragmentation of the spectrum (subdivision of the bands) when the generation number increases. An interesting result illustrated in Figure 3b-g is the occurrence of new characteristics around the frequency \(f_{c}=53\) MHz at the center of the gap associated with the periodic structure (Figure 3a). These resonances exhibit a certain order of repetition which is a typical characteristic of Fibonacci structures. This property, known as the scaling law [36; 64], has been considered as a sign of localization of the waves in Fibonacci systems. In the frame of a theoretical investigation of electronic transport in periodic and quasiperiodic mesoscopic structures with stubs and using the beautiful 'trace map' method introduced by [PERSON] et al. [36], [PERSON] et al. [64] demonstrated that the scaling behavior of the transmission spectra in Fibonacci stubbed structures is characterized by the scaling factor  
\[F=\sqrt{1+4(1+I)^{2}}+2(1+I), \tag{15}\]  
where \(I\) is an invariant that remains unchanged at each step of the recursive repetition [36; 64]. It was demonstrated that this invariant is given by [64]  
\[I=\frac{1}{4}[\cot(\frac{\omega d_{2}\sqrt{\epsilon}}{c})-\cot(\frac{\omega d_ {3}\sqrt{\epsilon}}{c})]^{2}\sin^{2}(\omega d_{1}\sqrt{\epsilon}/c). \tag{16}\]  
Similar expressions to Equations (15) and (16) have been derived by [PERSON] and [PERSON] [65] using Kohmoto-Kadanoff-Tang renormalization-group theory [79]. Furthermore, it has been demonstrated [36; 64] that the self-similarity property occurs around the frequency \(f_{c}=53\) MHz at the center of the gap of the periodic structure. This implies that the transmission coefficient should exhibit a self-similar behavior around the central frequency with \(T_{j+3}=T_{j}\) (i.e., the transmittance is periodic every three generations) and according to a scaling factor \(F\). For the frequency \(f_{c}\), Equations (15) and (16) give \(I=0.25\) and therefore \(F\simeq 5.2\). This phenomenon is clearly depicted in the insets of Figure 3b-g near \(f_{c}\). One can see a high similitude of the generations \(S_{4}\)-\(S_{7}\), \(S_{5}\)-\(S_{8}\) and \(S_{6}\)-\(S_{9}\) with a third order periodicity and a scaling factor \(F\) (see the inset around \(f_{c}\)) as it was found theoretically by [PERSON] et al. [64] in quantum wires with serial stubs and [PERSON] et al. [61] in Fibonacci serial loop structures. Nevertheless, it can be pointed out that due to the attenuation in the coaxial cables, the similarities between the different curves around the central gap frequency \(f_{c}\) do not exhibit exactly the same features as it is the case in the absence of loss (see the dashed curves in Figure 3). This is particularly pronounced for higher generations.