Retrieving method for elastic Willis metamaterials with mode couplings

In this work, a new method for retrieving effective material properties of Willis metamaterials is introduced. Although there have been several methods to retrieve effective parameters, they have focused on the case when only the Willis coupling or mode coupling exists, not the case when both couplings exist. Here, we developed a method that can be applied when both the Willis couplings and mode couplings exist at the same time. To retrieve effective parameters including both Willis and mode couplings, extremely complicated equations should be considered involving various parameters. To overcome the complexity, we utilized eigenvalues and eigenvectors of a transfer matrix from the scattering coefficients for the given structure. From the eigenvalues and eigenvectors, the effective material properties can be retrieved without complexity. We believe this new retrieving method can be an efficient tool to make various applications of Willis metamaterials.