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 in a way that allows us to draw conclusions that apply to a range of different metamaterial regimes. We consider the specific problem of single-frequency localisation, for which the appropriate effective model is a one-dimensional Schrodinger equation. Our results not only corroborate those of existing studies, but also extend them to other metamaterial regimes and provide a framework that allows us to compare many different gradient functions very efficiently. Hence, 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.