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Proceedings Paper

Scaling theory for homogenization of the Maxwell equations
Author(s): Alexei P. Vinogradov
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Paper Abstract

The wide application of composite materials is a distinctive feature of modern technologies. This encourages scientists dealing with radio physics and optics, to search for new type of artificial materials. Recently such investigations have shifted in the field of materials with weak spatial dispersion: chiral, omega materials, artificial magnets, etc. By weak spatial dispersion we mean that the constitutive relations are still local but constitutive parameters depend upon a wavenumber k. It is the dependence that is responsible for non-encountered-in-nature properties of the materials such as chirality [a first order in (ka) effect] or artificial magnetism [a second order in (ka effect)]. Here a is a typical size of an inclusion. Certainly, all these effects are small enough unless there is a resonance interaction of electromagnetic wave with an inclusion. Near the resonance frequency the effects are significant and perturbation theory in (ka) fails. Nevertheless it is convenient to describe the effects in terms of orders in (ka), understanding this as a matter of classification. In spite of physical clarity of the classification the constitutive relations are treated in terms of multipole expansion. The multipoles naturally appear at field expansion in (d/R) where d is the source size and R is the distance between the source and recorder. Such an expansion is useful in 'molecular optics' approximation where d very much less than r, with r to be a mean distance between the 'molecules.' Though the 'molecular optics' ceases to be a good approximation if we deal with composites where d approximately equals r, the mean current in the right hand side of the Maxwell equations is still expressed through multipoles (see Fig. 1). Below we consider the reasons justifying this sight on things even if we are working beyond the 'molecular optics' approximation. To repel an accusation in abstract contemplation let us consider examples of the 'multipole' media. Permeable composites made of non-permeable ingredients are well known. The simplest example is a composite loaded with highly conducting spherical inclusions. Due to eddy currents there appears a magnetic moment of the inclusion and the composite exhibits properties of diamagnetic. The inclusions of more complicated structure can exhibit resonant excitation resulting in induced magnetic moment. Examples of such inclusions are open rings, dielectric spheres, helix and bi-helix. In this case depending upon the relation between the working and resonant frequencies we can observe both diamagnetism or paramagnetism. Q-medium is more smart system. As the system of identical dielectric spheres is a permeable material, the system of different in size spheres may be non-permeable. The concentrations and radii may be chosen so that one part of spheres is excited in diamagnetic mode and the other in paramagnetic. Such a system is described by its quadrupole moment (see Fig. 1). Putting quantum mechanics apart we shall consider a classical composite material. The adjective 'classical' means that the scale of inhomogeneity is large enough to describe the reply of material on electromagnetic disturbance in terms of local constitutive equations Di equals (epsilon) ((omega) ,r)Ej ji equals (sigma) ((omega) ,r)Ej where (epsilon) ((omega) ,r), (sigma) ((omega) ,r) are local permittivity and conductivity.

Paper Details

Date Published: 14 November 1997
PDF: 12 pages
Proc. SPIE 3241, Smart Materials, Structures, and Integrated Systems, (14 November 1997); doi: 10.1117/12.293534
Show Author Affiliations
Alexei P. Vinogradov, Scientific Ctr. for Applied Problems in Electrodynamics (Russia)

Published in SPIE Proceedings Vol. 3241:
Smart Materials, Structures, and Integrated Systems
Alex Hariz; Vijay K. Varadan; Olaf Reinhold, Editor(s)

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