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Audible Download Audio Books. Alexa Actionable Analytics for the Web. On the other hand, it is obvious that medium dyadics satisfying a second-order equation define a certain class of media. It can be shown that media satisfying Equation. It is easy to see that if M satisfies Equation. Expressing Equation.

This means that, for a self-dual medium, there exist at most two distinct eigenvalues. The number of free medium parameters can be seen, from Equation. The class of self-dual media was previously introduced through D Gibbsian analysis in Ref. Because dyadics J appear to be similar to an imaginary unit, they are said to form an almost-complex structure in the space of medium dyadics M [,].

## - Document - Electromagnetic media with no dispersion equation

In fact, expressing Equation. As a special case of self-dual media let us consider the class of AB affine bianisotropic media, corresponding to the case when the dyadic T in Equation. The medium dyadic can then be expanded in the form [] T.

The AB medium is invariant in all spatial affine transformations. The eigenfields are defined by Equation. Since we know that there is no power propagation in a PEMC medium, the eigenfields cannot be power orthogonal. On the contrary, power propagation in a self-dual medium comes through an interaction of the eigenfields.

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## Electromagnetic media with no dispersion equation

In fact, considering the stress dyadic []. The significance of the class of self-dual media is in that certain transformations can be made for the sources and fields without changing the medium. This allows one to find a larger number of solutions for the field problem in that medium []. Further discussion on the properties of the transformation does not fall in the scope of this chapter. Certain classes of media could be defined in simpler terms when compared to their definition in classical Gibbsian vector analysis.

Main points in the analysis of fields in these media, treated more extensively in previous articles found in the list of references, were given and their connection to the corresponding studies using the classical Gibbsian analysis were briefly pointed out. Appendix: Multivectors, Multiforms, and Dyadics Application of differential forms in electromagnetic analysis requires some skill in using various identities in multivector and dyadic algebra.

In the literature the notation varies slightly from author to author. The present notation is based on that in Ref. Fields functions of space and time of dual vectors like the electric and magnetic fields E, H are called one-forms. Using the anticommutative exterior product other linear spaces are formed: E containing bivectors a i b i and F containing dual bivectors i i. More complicated products like a b c and give rise to multivectors and dual multivectors multiforms , respectively.

Dyadic algebra introduced by Gibbs as a coordinate-free representation of linear mappings in the D vector space [,] can be generalized to the algebra of differential forms, as was shown in Ref. Dyadics mapping vectors to vectors form a linear space denoted by E F. Other spaces are denoted similarly, e. Differential Forms and Electromagnetic Materials p-vectors over an n-dimensional vector space span a linear space whose dimension is n!

This can be continued to p-vectors and dual p-vectors. Because of the property Equation. The result is a scalar. Similarly, the incomplete double-duality products AB and BA can be defined between dyadics in certain spaces. In contrast, Gibbsian definition e i e i depends on the chosen basis. Because spaces F p and Fnp have the same dimension, dyadics F p Enp and Fnp E p mapping between these two spaces have special importance and they are called Hodge dyadics.

Also, dyadics mapping p-vectors to dual p-vectors and conversely are important and they are called generalized metric dyadics. In spaces E p E p and F p F p transpose of a dyadic is in the same space.

The sign denotes the scalar product between a p-vector and a dual p-vector while or denotes the contraction incomplete duality product between a p-vector and a dual q-vector []. Double products , , , follow laws similar to those defined in the Gibbsian dyadic algebra. The result is in the dyadic space E F.

Similarly we can define products A B C, etc. The double-wedge square of a dyadic A E F is. The unit dyadic mapping any bivector to itself is. More generally, the unit dyadic for p-vectors is I p. The unit dyadic mapping a p-form to itself is the transpose I p T. The inverse of a dyadic can be formed much in the same manner as in the Gibbsian D case [,]. Certain number of identities is essential in any analysis using differential forms because they reduce the need to expand expressions in terms of basis vectors and their basic relations.

Identities can be derived by expanding multivectors, dual multivectors, and dyadics in terms of basis vectors and dual basis vectors. Here we just give some examples taken from Ref. The result is another dyadic in the same space F E. A useful special case of Equation. Deschamps, Electromagnetics and differential forms, Proc.

IEEE, , , June. Hehl and Yu. Gibbs and E. Reprint of the edition. Lindell, Methods for Electromagnetic Field Analysis, nd edn. Lindell and A. Sihvola, Perfect electromagnetic conductor, J. Waves Appl. Antennas Propag. Lindell and K.

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Description This book applies the four-dimensional formalism with an extended toolbox of operation rules, allowing readers to define more general classes of electromagnetic media and to analyze EM waves that can exist in them End-of-chapter exercises Formalism allows readers to find novel classes of media Covers various properties of electromagnetic media in terms of which they can be set in different classes show more. Other books in this series.

Add to basket. Computational Methods for Electromagnetics Andrew F. From ER to E. Differential Forms in Electromagnetics Ismo V. Methods in Electromagnetic Wave Propagation D.