Waves in quasicrystals

Super band gaps in periodic approximants

The spectra of operators with quasi-periodic coefficients are notoriously hard to calculate or even characterise. A common approach to approximate their spectra is to use a supercell approximation whereby large piece of the material is used as the unit cell for a periodic approximant. Our recent work shows how the Floquet-Bloch spectra of periodic approximants faithfully reproduces the main spectral gaps of the quasi-crystalline operator [5]. In particular, we showed the existence of gaps that persist throughout a sequence of successively large periodic approximants (known as super band gaps) [4] as well as gaps that must persist for any periodic approximant due to the properties of the consituent elements (known as hierarchical band gaps) [3]. In both cases, these gaps will also exist in the spectrum of the limiting, non-periodic operator [5].

Symmetry-induced quasicrystalline waveguides

A common strategy for building a waveguide is to take a material with a spectral gap and add a defect (or series of defects) to create suitably localised eigenmodes. This is commonly done with periodic materials, however our recent work [1] provides a strategy for applying this principle in quasicrystalline materials by introducing an axis of reflectional symmetry. This artificial symmetry acts as a defect in the quasicrystalline structure and induces the desired wave localisation effect. This means the exotic spectral properties (which can include fractal collections of many, often large gaps) can be exploited for waveguide design.

Graded quasicrystals and fractal rainbow effects

Recent research published in Physical Review Letters [2] shows how quasicrystals can be spatially graded to produce a metamaterial that performs a new variant of the well-known "rainbow trapping" effect. The difference being that instead of modulating just one spectral gap (or a handful of gaps) as is normally done for rainbow trapping, a fractal collection of many spectral gaps is used. Since these gaps can also be very large, this leads to a broadband "fractal rainbow trapping" effect. Graded metamaterials are often used for energy harvesting applications and this approach provides a new way to potentially increase the operating bandwidth.