Evanescent Bloch waves in phononic crystals

Phononic crystals are two- or three-dimensional periodic structures that are composed with two or more materials with different elastic constants, giving rise to complete band gaps under specific conditions. Band structures are usually employed to describe infinite phononic crystals, as they provide one with all propagative waves in the periodic medium, or Bloch waves. It is however well known that evanescent waves must be considered in propagation problems whenever scattering, diffusion, or diffraction by a finite object are involved. We have extended the classical plane wave expansion (PWE) method so that it includes complex wave vectors in the direction of propagation at a fixed frequency. The new complex PWE method has been used to generate complex band structures for two-dimensional phononic crystals. Both propagative and evanescent solutions are found at once. This method of analysis is expected to become the basic building block to solve scattering problems in phononic crystals, yielding naturally diffraction eficiencies, as is illustrated with an example. In addition, it directly gives the eigenfrequency contours that are required to understand refraction (positive or negative) in phononic crystals.