## 1 Introduction  
After decades of extensive studies focusing on periodic crystals, scientific research interest have also been extended to quasi-periodic crystals. These structures were discovered by [PERSON] [1] in metallic Al-Mn alloys in 1984, for which he earned the Nobel Prize in Chemistry 2011. The samplesof such alloys exhibit a high degree of symmetry including the fivefold and icosahedral symmetries, which is prohibited by the rules of crystallography [1; 2; 3] and possess a long-range orientational order instead of a translational symmetry typical of a conventional crystal. The fascinating features of the quasiperiodic structures are that they have multiple frequency band gap regions similar to the band gaps of periodic crystals, due to the presence of more than one form of periodicity across these structures. Moreover, they exhibit transmission resonances and critical localized states [4]. The peculiar properties of quasi-crystals are strongly useful for the conception of potential applications in different domains [5; 6], especially optical communications technology such as optical filter [7; 8; 9], optical fibers [10; 11], microcavities and lasers [12; 13; 14; 15; 16], lenses [17], sensors [18; 19] etc. Studies of lower dimensional systems such as one dimensional (1D) layered media, waveguides and aperiodic linear chains are performed as in their 2D and 3D analogues with different applications [4; 5]. Indeed, it was shown that 1D models can be considered as a first approximation to the study of higher dimensional quasiperiodic systems [4]. Quasiperiodic structures could be structured following different types of sequences such as Fibonacci [20], Thue-Morse [21; 22], Rudin-Shapiro [23], Double period [24; 25] and Cantor [26; 27; 28]. The most well known of these systems is the Fibonacci structures [29], which have been widely studied for several types of excitations such as electrons [30; 31], phonons [32; 33; 34; 35], photons [36; 37], magnons [38] and plasmons [39; 40; 41]. [PERSON] et al. [30] produced first Fibonacci structures based on GaAs-AlAs semiconductor superlattices. Since this work, several works have been devoted to observe the exotic phenomena of Fibonacci systems [42; 43] and different interesting properties have been found [44] essentially by theoretical studies using simple 1D models. [PERSON] et al. [31] investigated the electronic features of a 1D quasi-periodic structure organized along a Fibonacci sequence such as the self-similarity and other critical properties of the wave functions. The first calculations of the phonon dispersion relations in a Fibonacci superlattice was developed by [PERSON] et al. [32] and a number of transmission dips associated with Bragg-like reflections of phonons has been found. In addition, a theoretical study for the propagation and localization of acoustic waves in a Fibonacci waveguide structure made of slender loop tubes connected together by slender tubes has been investigated by [PERSON] et al. [33] and experimentally checked by [PERSON] and [PERSON] [45]. [PERSON] et al. [36] have found that the transmission coefficient is multi-fractal and the resonance modes exhibit scaling behavior. The first experiment on photonic quasi-crystal based on the Fibonacci sequence has been reported by [PERSON] et al. [37]. It was shown that the phase spectrum exhibits clearly the self-similar structure of the dispersion curves typical of Fibonacci lattices. The research in Refs. [36; 37] have generated a large amount of theoretical and experimental research activity in the field of 1D, 2D and 3D photonic quasi-crystals with numerous device and engineering applications [5; 6; 46]. It was demonstrated that these systems exhibit fascinating properties like the occurrence of multiple forbidden frequency regions, self-similar spectrum and critical behavior of resonance modes, which can present an interesting alternative to ordinary photonic crystals for the design of new photonic devices [47; 48; 49; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59].