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Modelling of Dispersive Chiral Media: Limitations on Material Parameters
S. Tretyakov, E. Komarov
To cite this version:
S. Tretyakov, E. Komarov. Modelling of Dispersive Chiral Media: Limitations on Material Parameters.
Journal de Physique III, EDP Sciences, 1996, 6 (6), pp.721-725. �10.1051/jp3:1996151�. �jpa-00249486�
Short Communication
Modelling of Dispersive Chiral Media: Limitations on Material Parameters
S-A- llretyakov (*) and E.Yu. Komarov
St. Petersburg State Technical University, Radiophysics Department, 195251 St. Petersburg, Polytekhnicheskaya 29, Russia
(Received 18 December1995, accepted 11 April 1996)
PACS.41.90.+e Other topics in electromagnetism; electron and ion optics
PACS.75. Magnetic properties and materials
PACS.77. Dielectric, piezoelectrics and ferroelectrics and their properties
Abstract. Single-resonance models of the constitutive parameters of artificial isotropic chi- ral composites are analysed. Limitations on the medium parameters which follow from non-
negativeness of the field energy density are established and discussed. The analysis is restricted to the frequency regions where the losses
can be neglected. As a results, it is shown that the
single-resonance model cannot be used at frequencies far from the resonance or the low-frequency region.
1. Introduction
Complex media electromagnetics is a very rapidly developing area of applied physics. The interest in composite chiral materials at microwave frequencies is a clear sign of the potential prospects these media may offer, stimulating the fields of both materials science and electro-
magnetics. Anisotropic artificial materials with magnetoelectric field coupling (bi-anisotropic composites) can be synthesized so as to meet more specific requirements.
One of the most important problems in developing novel composite materials is the material parameters modelling, which should allow predictions of the material properties at operating frequencies from the geometry and concentration of inclusions. Here, care has to be taken in order to avoid non-physical parameter values in the models, and in this communication ~&~e
discuss some limitations on the models.
In particular, we deal with the theory of electromagnetic behaviour of artificial chiral com- posites designed for microwave applications. In the frequency domain, reciprocal isotropic
chiral composites can be modelled by bi-isotropic constitutive relations [11 D # EeffE j~efffi$H
B
# ~leIfH + j~elf@SE 11)
(*)Author for correspondence (e-mail: [email protected])
© Les #ditions de Physique 1996
722 JOURNAL DE PHYSIQUE III N°6
where Ee~r and ~le~r are the effective permittivity and permeability of the material, and Ke~r is the
chirality parameter (~). More general linear reciprocal bi-anisotropic media can be described
by the relations (see, e.g., [2,3])
D=I.E+h.H
B=$.H-fl E (2)
with dyadic parameters. Here T denotes the transpose operation, t and $ are symmetric dyadics.
Frequency behaviour of the effective media parameters I, $, K (or Ee~r, ~le~r, and ~e~r for isotropic mixtures) can be approximately predicted, using the Maxwell Gamett approach [4]
(see also [5] for treatment of isotropic composites). It allows to express the effective parameters of mixtures in terms of the dyadic polarizabilities t~ of single bi-anisotropic inclusions in local
fields Eioc, Hioc, defined as in
P " Tee Ejoc+fiem Hjoc (3)
m # Ume Eloc+Umm H]oc (4)
Here p, m stand for induced electric and magnetic dipole moments of individual inclusions.
The particle polarizabilities can be in turn estimated in the terms of the antenna model introduced in reference [6]. Within the frame of the uniaxial model and for low frequencies we
can approximately write [i, p. 241]:
~~~ ~~~~°~°
i ~J2LC + j~JCR~°~° ~~~
)mm = ammzozo "
~° ~~
zozo (6)
1- ~J ~~
+
~JCR
~~~ ~~~ ~~~~°~°
i ~J~~/~~JCR~°~° ~~~
Here the particles are assumed to be composed of a loop (loop area S) connected to a short wire dipole (length I) directed along the unit vector zo. The loop plane is orthogonal to zo.
Parameters of equivalent antennas (loop inductance L, dipole antenna capacitance C, the sum of the radiation resistances of the two parts R) can be found in antenna text books, e.g., [7].
~1 is the matrix permeability.
In the single-resonance model of the polarizabilities dispersion (5) ii), using the Maxwell Garnett formulas for isotropic low-density mixtures ii, Chapter 6], and neglecting losses, we arrive to:
ee~r=e+v ~
~
(8)
~Jo -~J
~2
tleR " i~ + "
2 2 19)
1d0 ld
~e~r=~ ~~°
~
(10)
~J~ ~J
(~) Here the vacuum parameters ED and /~o have been included to make the chirality parameter ~ea
dimensionless.
The resonance frequency ~J(
=
£, and the coefficients read:
n is the inclusion concentration, E and ~1 stand for the matrix parameters. The effective chirality parameter (10) coincides with that given by the Condon model [8].
Note an important relation between the polarizabilities ii, p. 241]:
aeeamm + aim
" o j12)
This relation reflects the particle symmetry: the loop is connected to the ~&~ire in its centre, and therefore the input currents of the two parts are always the same. This geometrical con- figuration is most effective in terms of maximizing magnetoelectric coupling, and for scatterers of other shapes or at higher frequencies the relation (12) does not hold.
2. Energy Density
In the following ~&~e establish physical limitations on the media parameters based on some
energy considerations.
In paper [9], the following condition for material parameters of lossless bi-aiiisotropic media
subject to equation (2) have been found: the dyadics
£ (Ldli+I S~~ lf)j
,
£ (Ldls+ lf t~~ 1)j l13)
must be positive definite. These restrictions were applied to the models of non-dispersive chiral and some bi-anisotropic media.
The low-frequency limit of these limitations in case of reciprocal bi-isotropic constitutive relations was considered in iii]. In that case we can assume that the permittivity and perme-
ability are constants with respect to frequency, and the chirality parameter is a linear function which tends to zero when frequency approaches zero.
In order to include in the analysis dispersive media models and establish more restrictive
relations, we employ an alternative expression for the energy density. Here we follow the proce- dure given in reference [10] for isotropic media and in reference ii ii for bi-isotropic materials.
Extending the analysis to bi-anisotropic constitutive relations (2), we arrive to the energy
density function W in the form
(u~i) dju~~*)
~ ~~~'~~ (~~~) ~~~l)
~~ ~~~~
duJ duJ
Here the asterisk denotes complex conjugate. For isotropic media described by equations (8)
to (10) the dyadic matrix in the above relation degrades to a conventional 2 x 2 matrix with scalar elements:
~~jj~~ Jfi~~ii~~
jis)
A
"
d(Q~N~ff) d(bJ~leff)
-j/S
~~ duJ
724 JOURNAL DE PHYSIQUE III N°6
These expressions for the energy density are meaningful for frequency areas with negligible absorption. This is the same restriction as in the simple case of isotropic media [10].
3. Parameters Limitations
The energy density of fields in passive media must be positive definite. Moreover, the difference between the energy density in matter and that of the same fields in free space must be also non-negative. From these statements, certain restrictions of physically admissible material
parameters follow.
Turning to the case of isotropic chiral media, we demand that the increment of the energy
density due to inclusions polarization must be positive definite. That means the matrix
A- ~ l(16)
l~
(where A is given by (15)) must be positive definite.
Demanding the trace and determinant of (16) to be non-negative and substituting the relations
(8) (10), we arrive to the restrictions
x § 3 (17)
4eollo'~J~x < (I + x)(3 x)x (18)
where x
= uJ~/uJ(.
In case if one uses the wire-and-loop model of chiral inclusions whose low frequency lumped- element description corresponds to the relations (5) (7), it is possible to establish very general
restrictive relations, independent from the particle size and inclusion density. Indeed, in that
case the model parameters satisfy
eol~o'~J~ = 1 (19)
(follows from (12)). Provided the equation (19) holds, the restriction (18) is satisfied only at two frequencies: in statics, when uJ
= 0 ix
= 0), and at the resonance, when uJ
= uJo ix
= I).
We emphasize again that the relation (19) corresponds to the maximum possible amount of
electromagnetic coupling for given particle dimensions, and therefore in reality the left-hand side of (19) is smaller than unity, and the restriction (18) is satisfied in certain areas around the two frequency points uJ
= 0 and uJ
= uJo.
Another inequality (17) is less restrictive and leads to the conclusion that the model (8) (10) cannot be used when frequencies uJ are higher than v5uJo.
4. Discussion
The above result means that the single-resonance models of complex-shaped inclusions polariz- abilities have natural limitations of their applicability areas. This limitation physically means
violation of the energy concept. The analysis has been restricted to the case of negligible losses,
~vhich is a severe limitation. Indeed, the most effective field coupling occurs in the resonance area, where the losses can be high. Moreover, all the dispersive media are also lossy, as follows from the causality principle. However, since the field energy density cannot be introduced as a
measurable physical quantity in dispersive lossy media, we have to confine the analysis. Any-
way, the conclusions appear to be important: single-resonance models inevitably loose sense at high frequencies, although such principles as the Kramers-Kronig relations can be satisfied.
As compared to the previous work on the field energy in bi-anisotropic media [9], the re-
strictions given here take into account dispersion effects and they are more strict since we demand the additional energy due to particle polarization to be positive. To do that, it seems easier to use the expression (14) for the energy density. Combining this approach with the relation between particle polarizabilities (12), very general limitations that do not depend on
the inclusion concentration or inclusion dimensions (until they are small enough) have been established. Using (14), the results can be extended to bi-anisotropic composite materials.
Acknowledgments
The research described in this publication was made possible in part by Grant 94-02-03453-a by
the Russian Foundation for Basic Research and Grant N°JI9100 from the International Science Foundation and Russian Government. The authors would like to thank Fr4dAric GuArin for suggestive discussions.
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