Abstract
We investigate both theoretically and experimentally the properties of electromagnetic waves propagation and localization in periodic and quasi-periodic stub structures of Fibonacci type. Each block constituting the Fibonacci sequence (FS) is composed of an horizontal segment and a vertical stub. The origin of the primary and secondary gaps shown in such systems is discussed. The behaviors and scattering properties of the electromagnetic modes are studied in two geometries, when the FS is inserted horizontally between two semi-infinite waveguides or grafted vertically along a guide. Typical properties of the Fibonacci systems such as the fragmentation of the frequency spectrum, the self-similarity following a scaling law are analyzed and discussed. It is found that certain modes inside these two geometries decrease according to a power law rather than an exponential law and the localization of these modes displays the property of self-similarity around the central gap frequency of the periodic structure where the quasi-periodicity is most effective. Also, the eigenmodes of the FS of different generation order are studied depending on the boundary conditions imposed on its extremities. It is shown that both geometries provide complementary information on the localization of the different modes inside the FS. In particular, in addition to bulk modes, some localized modes induced by both extremities of the system exhibit different behaviors depending on which surface they are localized. The theory is carried out using the Green’s function approach through an analysis of the dispersion relation, transmission coefficient and electric field distribution through such finite structures. The theoretical findings are in good agreement with the experimental results performed by measuring in the radio-frequency range the transmission along a waveguide in which the FS is inserted horizontally or grafted vertically.
Highlights
After decades of extensive studies focusing on periodic crystals, scientific research interest have been extended to quasi-periodic crystals
If we continue randomly substituting B blocks by A blocks (Figure 2f,g), one can notice that the central gap around 98 MHz becomes larger and the transmission dips associated to the secondary gaps in the transmission curves of the Fibonacci sequence (FS) disappear in order to create two separate allowed bands in the case of the system composed of only A blocks (Figure 2h)
Dashed curves correspond to the 6th (7th) generation. We have studied both theoretically and experimentally the behavior of the propagating and localization of electromagnetic modes in periodic and Fibonacci stub structures
Summary
After decades of extensive studies focusing on periodic crystals, scientific research interest have been extended to quasi-periodic crystals. The study of the FS in the two geometries (horizontal and vertical) enables us to deduce typical properties of these systems such as: (i) the fragmentation of the frequency spectrum into small bands and primary and secondary gaps, (ii) the self-similarity of the modes in the transmission spectra as well as in the electric field distribution around a central frequency where the quasi-periodicity is more effective, (iii) the analysis of the behavior of the eigenmodes of the FS with different boundary conditions imposed on its both extremities These modes are obtained by means of a theorem developed by some of us earlier [70,71] in which the eigenmodes of a finite system can be deduced directly from the maxima or minima of the transmission coefficient when this system is grafted vertically along a waveguide. A brief summary of the main results of this paper is presented in the conclusion
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