Complex Band Structure of Phononic Crystals and the Diffraction Problem

Abstract

The diffraction of elastic waves on a two-dimensional finite phononic crystal is investigated by a plane wave technique. It is first remarked that a full solution to the phononic crystal problem requires that all modes of the periodic structure (Bloch waves) are indentified and incorporated in the solution, including evanescent Bloch waves. An extended plane-wave expansion (PWE) method is used to obtain the complex band structure of the phononic crystal, but also the band structure of diffracted waves in the incident and exit media. Complex isofrequency curves are presented and show sharp variations of the Bloch wave vector with the angle of propagation. Finally, the complex band structures are used to formulate a reflection/transmission problem similar to the one leading to Fresnel formulas for homogeneous media. Some examples of diffracted field computations are given.