Tailoring phonon band structures with broken symmetry by shaping spatiotemporal modulations of stiffness in a one-dimensional elastic waveguide

Spatiotemporal modulations of the elastic properties of materials can be used to break time and parity symmetry of elastic waves. We show that the form of the elastic band structure depends not only on the spatial and temporal periodicity of a spatiotemporal modulation but also on its shape through its Fourier components. We demonstrate that hybridization gaps open from interactions between the Bloch modes of the periodic medium in absence of the temporal variation of the modulation and the combined sinusoidal components of the Fourier decomposition of the periodic modulation.