Anomalous Behavior of Heterogeneous Materials at Microwaves Frequencies: Introduction to Fractional Derivatives in Elect
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ANOMALOUS BEHAVIOR OF HETEROGENEOUS MATERIALS AT MICROWAVES FREQUENCIES: INTRODUCTION TO FRACTIONAL DERIVATIVES IN ELECTROMAGNETISM F. HELIODORE, D. COTTEVIEILLE AND A. LE MEHAUTE Laboratoires de Marcoussis, Centre de Recherches de Nozay, F 91 460 Marcoussis, FRANCE
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ABSTRACT The present note introduces new trends in electromagnetic spectroscopy in complex media. When an electromagnetic wave propagates in heterogeneous media, some questions arise about both physical meaning and validity range of the traditional analysis. The aim of our advanced research is related to the generalisation of Maxwell's equations able to describe both homogeneous and heterogeneous media from an unique point of view. In the frame of advanced electromagnetic research, the present note would like to introduce new trends in electromagnetic spectroscopy in complex media. As it is well known, the strict applicability of Maxwell's equations in their actual form is limited to homogeneous media, eventually separated by euclidean interfaces. In the case of an electromagnetic wave propagation (EMW) in heterogeneous media (especially when they are characterized by different geometrical scales), some questions arise about both physical meaning and validity range of these equations. A desirable aim of any advanced research would be the generalisation of Maxwell's equations able to describe both homogeneous and heterogeneous media in the frame of an unique point of view. As previously announced, this progress is surely able to be done if the media offer the regularity granted by fractal geometry [1). Moreover, it is well known that scattering analysis (X-ray, Neutron scattering, Optic scattering, ... ) is a suitable method to characterize heterogeneous materials. For instance, when a wave interacts with such a material, the scattering pattern may point out scaling properties related to the properties characterizing the eventual fractal character of the material (2,3,4). How can we consider interactions between an electromagnetic wave like a microwave and fractal materials? Can we characterize these materials using the usual parameters (E,gO)? Is the concept of electromagnetic impedance accurate? What is the meaning of the propagation constant in the medium? All these questions must be addressed in the frame of advanced research. In order to introduce the answer let us consider the traditional Maxwellian's approach based on the representation of the heterogeneous material as an effective medium. The properties are averaged on the volume and the behavior is supposed to be related to Maxwell's equations [5,6,7,8]. As a result we deduce effective parameters depending on frequency (and time) . The local average parameters, complex permittivity (S(C)), complex permeability (w(o)) and complex conductivity (c(03)) are considered as being representative of the behavior; these parameters are obtained by mixing laws (Maxwell-Garnett, mean field theory, energy balanced Mat. Res. Soc. Symp. Proc. Vol. 189. 01991 Materials Research Society
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