Evanescent Wave Boundary Layers in Metamaterials and Sidestepping them through a Variational Approach

TL;DR

This paper investigates evanescent boundary layers in metamaterials, clarifies their role at interfaces, and proposes a variational method to estimate scattering coefficients without explicitly modeling evanescent waves.

Contribution

It introduces a variational approach to approximate scattering at metamaterial interfaces, bypassing the need to explicitly compute evanescent modes.

Findings

The boundary layers are crucial for accurate interface modeling.

Ignoring boundary layers leads to discontinuities in displacement and stress.

The proposed method accurately estimates scattering coefficients beyond long wavelength limit.

Abstract

All metamaterial applications are based upon the idea that extreme material properties can be achieved through appropriate dynamic homogenization of composites. This homogenization is almost always done for infinite domains and the results are then applied to finite samples. This process ignores the evanescent waves which appear at the boundaries of such finite samples. In this paper we first clarify the emergence and purpose of these evanescent waves in a model problem consisting of an interface between a layered composite and a homogeneous medium. We show that these evanescent waves form boundary layers on either side of the interface beyond which the composite can be represented by appropriate infinite domain homogenized relations. We show that if one ignores the boundary layers then the displacement and stress fields are discontinuous across the interface. Therefore, the scattering coefficients at such an interface cannot be determined through the conventional continuity conditions involving only propagating modes. Here we propose an approximate variational approach for sidestepping these boundary layers. The aim is to determine the scattering coefficients without the knowledge of evanescent modes. Through various numerical examples we show that our technique gives very good estimates of the actual scattering coefficients beyond the long wavelength limit.