Basics of averaging of the Maxwell equations for bulk materials

• Published in: Metamaterials Vol. 6; no. 3-4; pp. 77 - 120
• Main Authors: Chipouline, A., Simovski, C., Tretyakov, S.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2012
• Subjects: Effective parameters; Electrodynamics; Equivalence; Homogenisation; Homogenization; Homogenizing; Macroscopic averaging; Mathematical models; Maxwell equation; Metamaterials; Wave propagation; Metamaterials; Macroscopic averaging; Homogenisation; Effective parameters
• ISSN: 1873-1988; 1878-0288
• DOI: 10.1016/j.metmat.2012.11.001

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 bulk MM, which is rather close to the case of compound materials but should include magnetic response of the inclusions and their clusters. In the vast majority of cases the consideration of bulk materials means that we consider propagation of an electromagnetic wave far from the interfaces, where the eigenwave in the medium has been already formed and stabilized. In other words, in this paper we consider the possible eigenmodes, which could exist in the equivalent homogenized media, and the necessary math apparatus for an adequate description of these waves. It has to be again clearly emphasized, that the presented paper does not suggest new recipes for the homogenization procedure, but rather summarizes known basics in order to establish solid basis for more particular cases. Nevertheless, it is believed that any homogenization model has to be compatible with the presented in this paper general structure. A discussion about boundary conditions and layered MM is a subject of separate publication and will be done elsewhere.