On the problem of comparing graded metamaterials

We use a simple effective model, obtained through the application of high-frequency homogenisation, to tackle the fundamental question of how the choice of gradient function affects the performance of a graded metamaterial. This approach provides a unified framework for comparing gradient functions efficiently and in a way that allows us to draw conclusions that apply to a range of different wave regimes. We consider the specific problem of single-frequency localisation, for which the appropriate effective model is a one-dimensional Schrodinger equation. Our analytic results both corroborate those of existing studies (which use either expensive full-field wave simulations or black-box numerical optimisation algorithms) and extend them to other metamaterial regimes. Based on our analysis, we are able to propose a design strategy for optimising monotonically graded metamaterials and offer an explanation for the lack of a universal optimal gradient function.