Basics of averaging of the Maxwell equations

TL;DR

This paper discusses the fundamental principles of averaging microscopic Maxwell equations to derive macroscopic equations, emphasizing the importance of consistency in homogenization models, especially for metamaterials with magnetic responses.

Contribution

It establishes basic principles for averaging Maxwell equations and applies them to metamaterials, ensuring models adhere to these principles for credibility.

Findings

Identifies fundamental principles for averaging Maxwell equations.

Proposes a consistent averaging procedure for metamaterials.

Highlights the importance of model credibility based on basic principles.

Abstract

Volume or statistical averaging of the microscopic Maxwell equations (MEs), i.e. transition from microscopic MEs to their macroscopic counterparts, is one of the main steps in electrodynamics of materials. In spite of the fundamental importance of the averaging procedure, it is quite rarely properly discussed in university courses and respective books; up to now there is no established consensus about how the averaging procedure has to be performed. In this paper we show that there are some basic principles for the averaging procedure (irrespective to what type of material is studied) which have to be satisfied. Any homogenization model has to be consistent with the basic principles. In case of absence of this correlation of a particular model with the basic principles the model could not be accepted as a credible one. Another goal of this paper is to establish the averaging procedure for metamaterials, which is rather close to the case of compound materials but should include magnetic response of the inclusions and their clusters. We start from the consideration of bulk materials, which means in the vast majority of cases that we consider propagation of an electromagnetic wave far from the interfaces, where the eigenwave in the medium has been already formed and stabilized. A basic structure for discussion about boundary conditions and layered metamaterials is a subject of separate publication and will be done elsewhere.