Analytic Materials

TL;DR

This paper develops the theory of inhomogeneous analytic materials with coefficients involving analytic functions, classifying them into types and exploring their properties, especially in two-dimensional cases, with implications for metamaterials.

Contribution

It introduces a classification of analytic materials based on the parameter p and extends the understanding of their properties, including special cases in two dimensions.

Findings

Complete analytic materials identified for maximum p

Progress in 2D analytic materials using rotation and potential concepts

Review of exact results and metamaterials for realizing desired coefficients

Abstract

The theory of inhomogeneous analytic materials is developed. These are materials where the coefficients entering the equations involve analytic functions. Three types of analytic materials are identified. The first two types involve an integer $p$. If $p$ takes its maximum value then we have a complete analytic material. Otherwise it is incomplete analytic material of rank $p$. For two-dimensional materials further progress can be made in the identification of analytic materials by using the well-known fact that a $90^\circ$ rotation applied to a divergence free field in a simply connected domain yields a curl-free field, and this can then be expressed as the gradient of a potential. Other exact results for the fields in inhomogeneous media are reviewed. Also reviewed is the subject of metamaterials, as these materials provide a way of realizing desirable coefficients in the equations.