Band structure of evanescent waves in phononic crystals

Evanescent waves must be considered in propagation problems whenever scattering, diffusion, or diffraction by a finite object are investigated. In the context of phononic crystals, they appear very naturally within frequency band gaps. Since no waves can propagate within a band gap, only evanescent waves are left to explain the exponentially-decreasing transmission of acoustic waves. We have extended the classical plane wave expansion (PWE) method so that it includes complex wave vectors in the direction of propagation. To do so, it is necessary to consider a fixed frequency and to solve for the wave vector, in contrast to the traditional way of obtaining band structures by considering any Bloch wave vector within the first Brillouin zone and solving for the frequency of allowed modes. The new complex PWE method has been used to generate band structures for two-dimensional silicon - void phononic crystals. Both propagative and evanescent solutions are found at once. The decay constants within band gaps are thus found and shown to depend on the wave polarization. The consideration of complex wave vectors also allows us to identify clearly the different branch systems in the band structure and to connect bands below and above band gaps. Furthermore, the distribution of the acoustic fields of evanescent modes can be computed. Their transformation from below to above a band gap as well as within the band gap itself is shown.