Complex Dispersion Relation Recovery from 2D Periodic Resonant Systems of Finite Size

The complex dispersion relations along the main symmetry directions of two-dimensional finite size periodic arrangements of resonant or non-resonant scatterers are recovered by using an extension of the SLaTCoW (Spatial LAplace Transform for COmplex Wavenumber) method. This method relies on the analysis of the spatial Laplace transform instead of the usual spatial Fourier transform of the measured wavefield in the frequency domain. We apply this method to finite dimension square periodic arrangements of both rigid and resonant scatterers embedded in air, i.e., to finite size sonic crystals and finite size acoustic metamaterials, respectively. The main hypothesis considered in this work is the mirror symmetry of the finite structure with respect to its median axis along the analyzed direction. However, we show that the method is robust enough to provide excellent results even if this hypothesis is not fully satisfied. Effectively, a minor asymmetry could be considered as a side effect when the structure is large enough because Laplace transforming the field along the main symmetry directions also implies averaging the field in the perpendicular one. The calculated complex dispersion relations are in excellent agreement with those obtained by an already validated technique, like the Extended Plane Wave Expansion (EPWE). The methodology employed in this work is intended to be directly used for the experimental characterization of real 2D periodic and resonant systems.