Plane wave expansion method for phononic crystals: review and prospects
Résumé
The plane wave expansion (PWE) method has been from the start at the heart of the development of the field of phononic crystals. It was for instance the theoretical tool used to support the seminal concepts put forward by Kushwaha et al. in 1993 [1]. At the start, the PWE method was limited to isotropic solid-solid compositions, but was then extended to fluid-fluid and solid-fluid compositions, with relative success only, to anisotropic materials, to piezoelectric materials, and to solid-void compositions. It has been employed for bulk acoustic waves, surface acoustic waves, and plate (slab) acoustic modes in periodically structured artificial materials. Today, the PWE method faces the strong concurrence of the finite-domain time-domain (FDTD) method, the finite element method (FEM), and the layer multiple scattering (LMS) method. Though it remains highly practical for generating the band structures of 1D and 2D perfectly periodic phononic crystals, it may suffer diverse convergence problems and is quite time consuming for 3D numerical simulations. Yet, has the PWE already given everything it had to offer? Recent progress in the representation of boundary conditions and of evanescent Bloch waves will be reviewed, and the use of the PWE for diffraction problems on finite size phononic crystals will be discussed. [1] M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys. Rev. Lett. 71, 2022 (1993).