3.2 Generalized Fibonacci sequence

1D Fibonacci quasiperiodic sequences are constructed by applying the inflation rule in [17]: σGFðH; LÞ : H ! HL; L ! H for the two blocks H and L, where H

Photonic Quasicrystals for Filtering Application DOI: http://dx.doi.org/10.5772/intechopen.81572


Table 1.

Repeated {H,L} blocks determined by applying the substitution rule σGTMðH; LÞ.

Figure 1.

Schematic drawing of 1D multilayered stacks made of dielectric (D)/superconducting materials (S), built according to the GTM(2, 2) sequence.

denotes the material with the higher refractive index, and L denotes the material with the lower refractive index. The GF chain is generated using the substitution rule: <sup>σ</sup>GFðH; <sup>L</sup><sup>Þ</sup> : <sup>H</sup> ! <sup>H</sup>mLn;<sup>L</sup> ! H. Thus, the GF sequence Sk+1 satisfies the recursion relation: Skþ<sup>1</sup> <sup>¼</sup> <sup>S</sup><sup>m</sup> <sup>k</sup> Sk n ‐<sup>1</sup> with <sup>k</sup> is the order of GF sequence.

The Fourier transform of Fibonacci class of quasicrystal gives discrete values which represent the significant property of crystals. We note that the eigenvalues of related matrix Fibonacci spectrum are Pisot numbers. For the Fibonacci-type, the material waves interfere constructively in appropriate length. The analysis of Fibonacci quasicrystals submitted to X-ray diffraction shows a multitude of Bragg peaks. Moreover, quasicrystals which are based on the Fibonacci distribution ordered at long distances, show a typical construction without a forbidden symmetry. Hence, the generalized Fibonacci (GF) type gives some basic proprieties which are identical to those given by simple Fibonacci class such as Fourier spectrum with Bragg peaks, inflation symmetry and localized critical modes with zero transmission called pseudo band gaps. In a generic form of the organized multilayers (H, L) through Fibonacci sequence, the four multilayered stacks are grouped in Table 2.

As an example, the third order of GF(m, n) quasiperiodic photonic structure containing alternate dielectric (D) and superconducting layers (S) with m = n = 2 is shown in Figure 2.


Table 2. Generation of Fibonacci sequence and organized blocks (H, L) repeated by the substitution rule σGFðH; LÞ.
