Fibonacci loop structures: bandgaps, power law, scaling law, confined and surface modes

We study theoretically and experimentally the existence and behavior of scaling law, power law, as well as confined and surface modes in one-dimensional (1D) quasiperiodic photonic bandgap structures. These structures are made of segments and loops arranged according to a Fibonacci sequence. We consider two types of structure, namely (i) a 1D periodic structure, called Fibonacci superlattice where each cell is a well defined Fibonacci generation. In these structures, we generalize a theoretical rule on the surface modes, namely when one considers two semiinfinite superlattices obtained by the cleavage of an infinite superlattice, there exists exactly one surface mode in each gap. This mode is localized on the surface either of one or the other semiinfinite superlattice. We discuss the existence of various types of surface modes and their spatial localization. The experimental observation of these modes is carried out in an original way, namely by measuring the transmission through a guide along which a finite superlattice (i.e., constituted of a finite number of quasiperiodic cells) is grafted vertically. The surface modes appear as maxima of the transmission spectrum; (ii) a Fibonacci sequence (FS) inserted horizontally between two waveguides. We give an experimental evidence of the scaling behavior of the amplitude and the phase of the transmission coefficient. In addition, by grafting vertically the FS along a guide, we obtain from the maxima of the transmission coefficient the eigenmodes of the finite structure with different boundary conditions. We show that both the finite sequence and the periodic Fibonacci structures exhibit the property of self similarity of order three at certain frequencies where the quasiperiodicity is most effective. The experiments are carried out by using coaxial cables in the frequency region of a few tens of MHz. These experiments are in good agreement with the theoretical model based on the formalism of the Green's function.