Photonic and phononic quasicrystals

Abstract

This review focuses on the peculiarities of quasiperiodic order for the properties of photonic and phononic (sonic) heterostructures. The most beneficial feature of quasiperiodicity is that it can combine perfectly ordered structures with purely point-diffractive spectra of arbitrarily high rotational symmetry. Both are prerequisites for the construction of isotropic band gap composites, in particular from materials with low index contrast, which are required for numerous applications. Another interesting property of quasiperiodic structures is their scaling symmetry, which may be exploited to create spectral gaps in the sub-wavelength regime. This review covers structure/property relationships of heterostructures based on one-dimensional (1D) substitutional sequences such as the Fibonacci, Thue–Morse, period-doubling, Rudin–Shapiro and Cantor sequence as well as on 1D modulated structures, further on 2D tilings with 8-, 10-, 12- and 14-fold symmetry as well as on the pinwheel tiling, the Sierpinski gasket and on curvilinear tilings and, finally, on the 3D icosahedral Penrose tiling.

Low-contrast bandgaps of a planar parabolic spiral lattice

“…In the high ⌬⑀ regime, the strong scattering leads to gaps based on Bragg reflection from the locally distorted four-or sixfold ordering. These gaps, which correlate with those of more conventional structures, can also be viewed in terms of the Mie resonance condition for single rods [3]. For low ⌬⑀, the long-range interaction permits sampling of the sunflower's unique spiral structure.…”

“…Many of their potential applications require a complete photonic bandgap (PBG), which is more readily achieved in lattices with high rotational symmetry. Aperiodic crystals, or quasi-crystals, may possess arbitrarily high local or statistical symmetry, suggesting wider isotropic gaps in weakly modulated ͑⌬⑀ Ͻ 3͒ structures [2,3].…”

“…Feedback therefore occurs isotropically from the Bragg ring, which incorporates many spatial frequencies to further broaden the gap. In contrast, Bragg reflection from periodic crystals acts on a sparse pure-point Fourier space [3], thereby restricting the overlap of low-contrast spectral gaps.…”

Lempel-Ziv Complexity of Photonic Quasicrystals

“…Deterministic aperiodic spatial patterns, dubbed photonic quasicrystals [14][15][16][17][18][19][20][21], display stunning optical responses that cannot be found in either periodic or random systems. Among other issues, they include a self-similar energy spectrum [22,23], a pseudo-bandgap of forbidden frequencies [24][25][26], and critically localized states [27,28] whose wave functions are distinguished by power law asymptotes and self-similarity [29][30][31].…”

Guiding Properties of a Photonic Quasi-Crystal Fiber Based on the Thue–Morse Sequence

“…However, the concept of order without periodicity has been studied extensively in science and technology [10]. In fact, some interesting review articles on photonic quasicrystals can be found in the literature [11,12]. It has been demonstrated that photonic quasicrystals can also exhibit photonic band gaps [13][14][15].…”