Dispersion relation of coupled-resonator acoustic waveguides formed by defect cavities in a phononic crystal

We investigate the dispersion of phononic crystal waveguides formed by evanescent coupling of a chain of defect cavities and supporting slow-wave propagation. These coupled-resonator acoustic waveguides (CRAWs) are analogous to the coupled-resonator optical waveguides formed in photonic crystals. CRAW dispersion can be controlled by increasing the distance between cavities, with the result of decreasing their coupling, and hence flattening the dispersion relation. Based on the tight-binding model, the dispersion relation is found in the form of a Fourier series expansion with explicitly given coefficients. This model is tested against the exact dispersion relation of a two-dimensional solid-solid phononic crystal of tungsten inclusions in a silicon matrix and only partial agreement is found. An alternative model of a linear chain of coupled resonators, resting only on the hypotheses of linearity and periodicity, is then proposed. While the Fourier coefficients in this model are a priori unspecified, they can be fitted against the exact dispersion relation, resulting in an excellent agreement with only a few terms in the Fourier series expansion. The Fourier coefficients are shown to be a direct measure of the coupling of neighbouring resonators.