between the electric- and magnetic-field vectors (E and H) and the electric- and magnetic-flux densities (D and B). The three scalar material parameters are electric permittivity ( E ), magnetic permeability (,U), and the magnetoelectric Tellegen parameter ( a ). Now, return to the claim of [ l ] that this Tellegen medium is indistinguishable from ordinary dielectric-magnetic medium. This claim is clearly wrong, because a dielectric-magnetic medium (which only possesses permittivity E and permeability p ) would respond to electric excitation like a simple dielectric, i.e., with only electric polarization and no magnetic polarization, which is a purely electric response. Therefore, a Tellegen medium cnn be distinguished from a dielectric-magnetic medium by measuring, for example, its response to plain electric excitation. if there is masnetic polarization, it is a Tellegen medium; if not, it is an ordinary medium. And, of course, the same test to distinguish between the materials can be made using a static external magnetic field [6].
To conclude, a consequence of the theory expounded in [ I ] does not hold It is a serious matter If a theory leads to a false conclusion, this theory is wrong, according to the rules of classical logic References
1 W S Weiglhofer and A Lakhtakia, “A brief review of a new development for constitutive relations of linear bi-anisotropic media,” IEEE A n t m w m niid Propngnfron Mngnzrr?e, 37, 3, June 1995, pp 32-35
2. E. J. Post, Formal S‘tr~iicfi~i~ of Elecfroningiefics, Amsterdam, North-Holland, 1962. The result Weiglhofer and Lakhtakia base their analysis on is Equation (6.18) on page 129. Note that Post himself gives up this result in his later study, “The Constitutive Map and Some of its Ramifications,” A11rinls of Physics, 71, 1972, on page 505. 3. I. V. Lindell, A. H Sihvola, S. A. Tretyakov, and A. J. Viitanen, Hectiaiiingnetic Wnwr in Chiml nnd Bi-isofropic Medin, Boston and London, Artech House, 1994. The label “Tellegen”’ gives credit to the original article by B. D. F. Tellegen: “The Gyrator, a New Electric Network Element,” Philips Research Reports, 3, 2, 1948, pp. 81-101. 4. A. Sihvola, “Are Nonreciprocal Bi-isotropic Materials Forbidden Indeed?”, IEEE Transacfioi7s on Microwave Theory and Techniqzies, September 1995, to appear; A. Sihvola, “When Doubting Tellegen Material Give Her the Benefit of the Doubt,” Chiral discussion forum “CHIRAL-L” at
[email protected] alias
[email protected], 21 December 1994; A. Sihvola, “In Defence of Isotropic Magnetoelectric Media on the Bi-Anisotropic Electromagnetic Battlefield,” Pyoceedings of /he I 9 9 5 I~term7tionnl Syniposiiini on Electroningmtic Theory (URSI), May 23-26, 1995, St. Petersburg, Russia, pp. 8-10. 5. It is essential that all dipole pairs are similar in their sense of coupling the electric- and magnetic-dipole arrows. In the example of Figure 1, the positive end of the electric dipole is assigned with the north pole of the magnetic dipole. The opposite choice would still be magnetoelectric, but the sign of the a parameter would change in the constitutive relations.
orient themselves. But we always have time to wait for the response to build up, using a static (or low-frequency) external field. And, the response which remains after the transients have faded away is magnetoelectric for Tellegen medium.
Ari Sihvola Electromagnetics Laboratory Helsinki University of Technology 021 50 Espoo, Finland Tel: (358-0) 4512261 Fax: (358-0) 4512267 E-mail: ari. sihvola@hut . fi
1. Abstract We show that Sihvola’s counter-claim against a certain constraint on linear constitutive relations is erroneous. Pending the production of an actual sample of a non-reciprocal bi-isotropic medium, we opine that the constraint remains physically valid, besides being mathematically proper in the context of modern electromagnetic theory.
2. Introduction
r. Ari Sihvola wrote his comments “to point out that not everyone in the electromagnetics community agrees with results of [I].” We have no illusions to be dispelled thus. Nevertheless, we are grateful to him for providing us with yet another opportunity for “actively promoting” our mathematical findings. We have chosen to reply at length in order to (1) reiterate several issues that Sihvola keeps on ignoring, despite repeated clarification, as well as to (2) clarify the new confusion he has created by his comments. The basis of modern electromagnetic theory is the Lorentz force, which proclaims E(x,f) and B(x,f) as the only electromagnetic fields in free space. Matter is represented by a collection of discrete charges. In any material medium, after suitable averaging, the induction fields D(x,t) and H(x,t) arise at macroscopic-length scales. Constitutive relations must be prescribed for relating the induction fields {D(x,t),H(x,f)} to the primitive fields
{E(x,t),B(x,t)} in any material medium. The construction of these relations is primarily phenomenological, though certain epistemologically mandated proprieties must be adhered to. In particular, constitutive relations of a certain medium must be unique with respect to the Maxwell equations. The constitutive relations appropriate for a linear, spatially local, homogeneous, causal medium may be stated as
6. Weiglhofer and Lakhtakia seem to worry about media with instantaneous response in [ l ] The medium discussed here is certainly not instantaneous, because it takes time for the dipole pairs to
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IEEEAntennas and Propagation Magazine, Vol. 37, No. 5,October 1995
where the tensor 2 ( 1 ) is the dielectric-susceptibility operator, -e are called the magnetoelectric operators. All four operators are causal (i.e., 2 ( t ) = g for f < 0 , etc.), because all materials must
claimed that if a non-reciprocal bi-isotropic (NRBI) “material could exist, it would be indistinguishable from a simple dielectric-magnetic substance.” We did not make that claim. Although Sihvola may not distinguish between Tellegen media and NRBI media, we do.
exhibit delayed response; see Appendix A of [ 11 for hrther details. We have also assumed in Equations (la, lb) that the response properties of the medium do not vary with time.
Let us clarify the situation for the reader’s benefit. Suppose a general bi-isotropic medium, described by a specialization of Equations (la, lb), exists. Its constitutive relations may be stated as
Insisting on general covariance, Post [2] determined that the constraint
D(x, t) = @(x,
2 ( t ) is the magnetic-susceptibility operator, while g ( t ) and p(t) =
=ni
=e
must be satisfied, this relation being the equivalent of Equation (6.18) of [2] in a different notation. Incidentally, Post does not appear to have explored its physical consequences; thus, Post’s equation is more a starting point for our work than its base, contrary to Sihvola’s suggestion. Last year, we gave an alternate proof [3] of Equation (2): When Equations (la, Ib) are substituted into the Ampere-Maxwell equation, V x H(x,f)-2,D(x,f) = j ( x , t ) , and the induction fields are eliminated, a redundancy emerges with respect to the Faraday equation, V x E(x,t)+alB(x,t)= 0 , The same redundancy emerges the
divergence
+ E” 6 [ Z e ( 7 ) *E(x, t - r)]dr
+
Trace[g(t)-z(t)] = 0
between
1)
equations
[[G(
(:
7 )-
T)]
B(x ,t - 7)Id.r
X,~ H(x,t)= - 1B (- X , ~ ) - -1[ ~ ~ [ ~ ~ , ( ~ ) - B-(.r)]dz
PO
Po
+joy[ (. T ) +(:
T ) ] E(x, t
-
The distinctions between NEW, Tellegen, and chiral media [9] become clear from the following table:
V * D(x,f) = p ( x , t ) and
The elimination of this redundancy, as demanded by the principle of parsimony, provides the alternate proof of Equation (2). Later, we proved that an appropriate modification of Equation (2) holds for nonhomogeneous media [4]. Our latest proof is for spatiotemporally nonlocal media [5], which extends Equation (2) to all linear media. V*E(x,t) = 0.
Sihvola did not find fault with our mathematical results, and neither did the reviewers of our relevant papers. Instead, Sihvola complained that Equation (2) leads to “loss of physics.” We completely disagree with him, as we will now show.
-m
