Graded quasiperiodic metamaterials perform fractal rainbow trapping

Abstract

The rainbow trapping phenomenon of graded metamaterials can be combined with the fractal spectra of quasiperiodic waveguides to give a metamaterial that performs fractal rainbow trapping. This is achieved through a graded cut-and-project algorithm that yields a projected geometry for which the effective projection angle is graded along its length. As a result, the fractal structure of local band gaps varies with position, leading to broadband fractal rainbow trapping. We demonstrate this principle by designing an acoustic waveguide, which is characterised using theory, simulation and experiments.

Figures (10)

• Figure 1: Experimental schematic and cut-and-project algorithm: (a) 127 acoustically rigid rods (of diameter 4 mm) are placed at positions determined by the graded cut-and-project algorithm, forming an array of length $L$ = 1.5 m, shown without the enclosing waveguide. (b,c) End-views of source (loudspeaker) and receiver (microphone) positions respectively, with enclosing waveguide shown with a square-cross section of width $w$ = 2 cm. (d) Schematic of the graded cut-and-project algorithm, which projects a square lattice $\Lambda$ onto a quadratic curve. The rod positions and sample geometry are further detailed in the supplementary material.
• Figure 2: The spectral 'butterfly'. Varying the cut-and-project angle $\theta$ of a square lattice causes the band gaps to shift and open or close. Any points in the square periodic lattice $\Lambda$ that are within a distance $w$ of the straight line $\Gamma$ are projected onto the line, yielding the quasicrystal $\mathcal{P}(\Lambda)$.
• Figure 3: Comparison of the transmission spectra between (a) theory and lossless FEM simulations, and (b) Lossy FEM simulation (thermo-viscous physics) and experimental results. Note the same scales in both panels.
• Figure 4: Demonstration of the fractal rainbow effect. FEM simulations showing frequency spectra as a function of position. The numbered lines show the measurement positions (supplementary material), with the corresponding experimental comparisons shown in the numbered plots (same frequency axis).
• Figure 5: If the projection angle $\theta$ is such that $\tan\theta$ is rational, then the projected quasicrystal is periodic so its spectrum can be computed using Floquet-Bloch.