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Crystals 2016, 6(8), 93; doi:10.3390/cryst6080093

An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces

1
Department of Electrical Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran 15875, Iran
2
Department of Electrical Engineering, The Pennsylvania State University, University Park, PA 16802, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Enrique Maciá Barber
Received: 18 May 2016 / Revised: 31 July 2016 / Accepted: 4 August 2016 / Published: 10 August 2016
(This article belongs to the Special Issue Structure and Properties of Quasicrystals 2016)
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Abstract

A rigorous method for obtaining the diffraction patterns of quasicrystals is presented. Diffraction patterns are an essential analytical tool in the study of quasicrystals, since they can be used to determine their photonic resonances. Previous methods for approximating the diffraction patterns of quasicrystals have relied on evaluating the Fourier transform of finite-sized super-lattices. Our approach, on the other hand, is exact in the sense that it is based on a technique that embeds quasicrystals into higher dimensional periodic hyper-lattices, thereby completely capturing the properties of the infinite structure. The periodicity of the unit cell in the higher dimensional space can be exploited to obtain the Fourier series expansion in closed-form of the corresponding atomic surfaces. The utility of the method is demonstrated by applying it to one-dimensional Fibonacci and two-dimensional Penrose quasicrystals. The results are verified by comparing them to those obtained by using the conventional super-lattice method. It is shown that the conventional super-cell approach can lead to inaccurate results due to the continuous nature of the Fourier transform, since quasicrystals have a discrete spectrum, whereas the approach introduced in this paper generates discrete Fourier harmonics. Furthermore, the conventional approach requires very large super-cells and high-resolution sampling of the reciprocal space in order to produce accurate results leading to a very large computational burden, whereas the proposed method generates accurate results with a relatively small number of terms. Finally, we propose how this approach can be generalized from the vertex model, which assumes identical particles at all vertices, to a more realistic case where the quasicrystal is composed of different atoms. View Full-Text
Keywords: quasicrystals; diffraction pattern; higher dimensional approach; cut-and-project quasicrystals; diffraction pattern; higher dimensional approach; cut-and-project
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Namin, F.A.; Werner, D.H. An Exact Method to Determine the Photonic Resonances of Quasicrystals Based on Discrete Fourier Harmonics of Higher-Dimensional Atomic Surfaces. Crystals 2016, 6, 93.

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