Spectral Properties of 1D Fibonacci Quasicrystals

This Demonstration shows the spectral properties of the Fibonacci quasicrystal from both algebraic and graphical points of view; in the latter case in real and reciprocal space. The Fibonacci quasicrystal is the most studied one-dimensional (1D) quasiperiodic model structure, given its interesting applications in photonics and acoustics.
Like all other 1D quasiperiodic structures discussed in [1], the construction of such a system is based on a substitution sequence defined by a two-letter alphabet and a substitution rule , that can be expressed in matrix form as
,
using the substitution matrix
.
This procedure can be iterated times; it corresponds to the generation of a single periodic approximant for the 1D quasicrystal system, which can be replicated times to give an adequate approximation (patch) for an infinite quasicrystal [2].
This Demonstration implements the two-letter word generation based on such a rule up to a finite number of nesting substitutions using the appropriate controls. For the Fibonacci substitution matrix, the computed eigenvalues are the golden ratio
and its negative reciprocal
.
It can be easily checked that at each nesting iteration the frequencies of the letters and are consecutive Fibonacci numbers. As tends to infinity, the ratio of the frequencies of to tends to , since the ratio of two consecutive Fibonacci numbers tends to the golden ratio .
The frequencies of the two letters and at each nesting iteration have been computed in two reciprocally consistent ways, by simply counting the number of occurrences of the letters and in the generated letter sequence and by implementing the following power matrix relation:
.
By selecting the "plots" setter, you can see the resulting 1D spectrum in real space for a single periodic approximant (impedance plot), where and are assigned to spacings of lengths and 1, respectively. The distance from the origin is normalized by the period of the letter sequence (or periodic average structure) , where is set to 1, in accordance with [1]. A bar of unit length 1 is put at each coordinate point in order to give a pictorial idea of the quasiperiodicity of the system.
You can visualize the reciprocal space representation in Fourier space by selecting either the "Fourier" setter for the corresponding transformed power spectrum or the "transmission" setter for the corresponding transmittance spectrum on a logarithmic scale. The slider control "number of approximants " lets you obtain a denser sampling of reciprocal space points in order to see higher-order peaks in the Fourier spectrum. This is achieved numerically by replicating the fundamental letter sequence used to construct the single periodic approximant times. The computed Fourier power spectrum can also be considered as a simulation of the diffraction pattern of the 1D Fibonacci quasicrystal.

SNAPSHOTS

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DETAILS

Snapshot 1: letter sequence for the single periodic approximant at the initial stage with , with a frequency ratio limit still far from the golden number
Snapshot 2: same as the Thumbnail image but with transmission plot (, ) enabled, instead of the Fourier power spectrum
Snapshot 3: Fourier spectrum similar to the Thumbnail image but with a higher value, therefore with more spectral components
References
[1] W. Steurer and D. Sutter-Widmer, "Photonic and Phononic Quasicrystals," Journal of Physics D: Applied Physics, 40(13), 2007 R229. doi:10.1088/0022-3727/40/13/R01.
[2] U. Grimm and M. Schreiber, "Energy Spectra and Eigenstates of Quasiperiodic Tight-Binding Hamiltonians," Quasicrystals: Structure and Physical Properties (H.-R. Trebin, ed.), Weinheim, Germany: Wiley-VCH, 2003 pp. 210–235. arxiv.org/abs/cond-mat/0212140.
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