HAL Id: jpa-00249286
https://hal.archives-ouvertes.fr/jpa-00249286
Submitted on 1 Jan 1994
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Enhanced polarization effects in chiral composites
A. Sihvola
To cite this version:
A. Sihvola. Enhanced polarization effects in chiral composites. Journal de Physique III, EDP Sciences, 1994, 4 (12), pp.2601-2607. �10.1051/jp3:1994107�. �jpa-00249286�
Classification Physics Abstracts
77.30 78.20E
Enhanced polarization effects in chiral composites
A. H. Sihvola
Helsinki University of Technology, Electromagnetics Laboratory, Otakaari 5 A, 02150Espoo,
Finland
(Received 29 Marc-h J994, accepted J5 September J994)
Abstract. This communication predicts an enhanced polarization phenomenon (ECP, enhanced chiral polarization) in mixtures of chiral materials. Chiral, or handed, media are of increasing
interest in microwave technology these days. ECP is shown to appear for chiral inclusions of flat shape, provided that the chirality parameter of the inclusions approaches the value of the refractive
index of the sample. The report also compares ECP with the dielectric anomaly, observed with metal mixtures.
1. Introduction.
Microwave engineers and scientists have recently become aware of the potential applications
of chiral materials. Consequently, efforts are made to produce material samples with desired electric, magnetic, and chiral parameters. In several research groups in the United States, Westem Europe, and Russia, materials have been manufactured that display notable chiral
activity at microwave and millimeter wave frequencies I]. These media have been tailored by mixing handed elements in a host material. The elements are often spirals made of metal or dielectric. The possible applications of these materials range from polarization-correcting
radomes and lenses through microwave couplers to radar cross section reduction.
Chirality is a geometric concept, chiral medium being either left-handed or right-handed [2].
As far as the interaction of material with electromagnetic waves is concerned, chirality brings
forth magnetoelectric coupling electric field creates not only electric polarization but also magnetic, and vice versa. On the level of constitutive relations, in SI units, the chiral medium
behaves according to the following [3]
b
=
Et j« fill (I)
B
= ~H + j« fiE. (2)
These give the magnetoelectric relation between the electric # and magnetic fields
H, and electric D and magnetic flux densities fi. In addition to perniittivity
e and permeability
@Les Editions de Physique 1994
2602 JOURNAL DE PHYSIQUE III N° 12
~, there appears a dimensionless cross-coupling term K, which is the chirality parameter. j is the imaginary unit, indicating the time-harmonic dependence expowt) and ~o, so are the
permeability and the permittivity of free space. Often this type of reciprocally chiral medium is called « Pasteur medium » to honor the French scientist who, in the 1840's, discovered the
connection of handedness to optical activity.
It is the purpose of this paper to describe some qualitatively new aspects that appear in the wave-material interaction as there is chiral magnetoelectric coupling in the medium. The effective properties of heterogeneous media are often complicated functions of the electroma-
gnetic parameters of the component phases, and also of the microscopic geometry of the
material. These laws have to be known as composites are being designed. Due to the
crosscoupling, in chiral composite modeling the dielectric design is not independent of
magnetic design but affected by it. The amount of crossdependence is proportional to the
chirality parameter K.
This coupling can be seen in the simplest polarizability coefficients of chiral particles which
are functions of all material and shape parameters of the inclusion. It can be seen in the counterintuitive fact that components with no magnetic properties (I.e. the permeabilities of all components being those of three space, ~ = ~o), mixed together, will create a medium with
nonzero magnetic susceptibility [3, 4]. It is even possible to tailor diamagnetic mixtures using
only paramagnetic components [5].
But the chiral coupling can also in some cases produce unexpectedly strong polarization in both electric and magnetic domains. This, however, requires a rather strong magnitude of the
chirality parameter. Furthermore, the enhancement depends on the shape of the chiral inclusion in a sensitive manner. In the following, a quantitative treatment of this phenomenon,
which I call here Enhanced Chira/ Polarization (ECP), will be performed. Due to the fact that the electromagnetic modeling of chiral materials is a very recent topic of study, it seems that
ECP has neither been experimentally observed, nor even theoretically predicted.
2. Polarization modeling of chiral heterogeneity.
As one describes matter as a collection of discrete inclusions immersed in a continuous
background medium, the macroscopic modeling requires knowledge about the polarizability components of the inclusions. Polarizability is the relation between the incident electric and
magnetic field. E, H and the (electric and magnetic) dipole moments fi~, fi~ induced in the
scatterer. Due to the magnetoelectric coupling, the polarizability of a chiral inclusion is a
2 x 2 matrix
~~
1= "~~ "~~ ~ (3)
p~ ~ me ~
mm H
with the copolarizability a,, and crosspolarizability components a,~.
2.I POLARIzABILITtES. Solving the polarizability components of a chiral particle of
arbitrary shape is equivalent to determining the fields excited by the inclusion as it is immersed in electromagnetic field. In analytic form the problem can be solved for ellipsoidal shapes. The
polarizability matrix of a chiral ellipsoid with material parameters s, ~, K has been solved [6].
The polarizability components in equation (3) are dyadic (or tensors) rather than scalars,
because an ellipsoid is not spherically symmetric. Using the unit vectors fi, along the three axes
3
of the ellipsoid, the polarizability dyadics &~~
= jj a~~
,
fi, fi, are
,
so V
"ee, f (~ ~0)i~, f~ + (1 fi~,)f~0j fi~, K~1~0 ~0) (4)
,
«mm., = ~],~ l(~ ~o)iN, ~ + (i N, ) eui N, K2 ~u Eo) (5)
«~~,, = «~~, =
~J ~[° ~° ~ i
~~~
with ,
A, = IN, ~ + (i N, ) ~oj IN, e + (i N, ) sol N) K2
~o So (7)
and V
= 4 grabc/3 being the volume of the ellipsoid. a, b, c are the semiaxes of the ellipsoid,
and the depolarization factors are [7]
~~ ~~
i~
(s + a~) ~/(s
+
~)(s
+ b~)(s + c~)
~~~
For depolarization factors N~ and N~, interchange b and a, and c and a in equation (8), respectively. The depolarization factors satisfy N~ + N~ + N,
= I for any ellipsoid, and for a sphere, there are equal N~ =N~ =N~ =1/3. The other two special cases are a disk
(depolarization factors1, 0, 0) and a needle (0, 1/2, 1/2).
As a special case of spherical particle, for which the three depolarization factors are equal,
we see that also the x, y, and z components of the polarizabilities become equal. These
polarizability dyadics therefore reduce to four scalars
(s Eo)(~ + 2 ~o) K ~ ~o so
a~~ = 3 so V (9)
(JL + 2 l~o)(E + 2 So) K~ l~o SO (l~ l~o)(e + 2 so) K ~ jlo so
"~~ " ~ ~° ~
(~ + 2 ~o)(s + 2 ~u) ~ 2 ~u so ~~°~
9 JK/l~ F~ V fi
~~~ ~~~
(/~ + 2 mu)(p + 2 so)
K
2 ~u So
~~ ~~
which expressions are indeed the correct result [8].
2.2 EFFECTIVE-MEDIUM DESCRIPTION. Looking for strong polarization effects in media,
I-e., very large effective material parameters for a composite, one needs to use inclusions with maximized polarizability expressions. The maximum of expressions (4-6) can be found if the denominator A, can reach the value zero. In the following, this condition will be called Enhanced Chiral Polarization (ECP).
However, there are limitations for the material parameters of a sample of chiral medium.
The chirality parameter « cannot be arbitrary large. The following inequality can be written :
K w n = l~~ (12)
l~o So
over which there exists a rather unanimous consensus ill. The restriction (12) prohibits ECP to appear in connection with spherical chiral inclusions, since the denominator always has a
positive value, being a multiple of (s + 2 so)(~ + 2 ~o) K? ~o so.
2604 JOURNAL DE PHYSIQUE III N° 12
However, the denominator can be made to vanish within the allowed limits (12). This
happens by putting N,
= I, whence the denominator becomes
A, = ~e K 2~o so (13)
which disappears just at the upper limit for the chirality parameter K. This case corresponds to a
flat ellipsoid, where one of the axes is much smaller than the other two, for example a disk.
And ECP arises only for one field direction, which is the direction normal to the flat surface of
the inclusion in the other two directions, the polarizability components correspond to
ordinary media, which are electrically and magnetically polarizable in the normal manner.
In terms of the polarizability components, the effective material parameters can be written if
one knows the number density n of the inclusions in the mixture. The simplest case is the Maxwell Gamett mixing formula for chiral inclusions immersed in isotropic nonpolarizable background material (permittivity so, permeability ~o). For dilute mixtures (the chiral inclusions occupy a small fractional volume), the expression for the effective material parameters of the mixture s~~~, ~~~~, K~~~ is straightforward.
If the chiral ellipsoids are all aligned, the mixture is anisotropic (to be more accurate, it is
bianisotropic), with the parameter components
~etf,, " ~0 + ~~ee, (14)
iLeff,, " 1L0 + ntYmm,, (15)
K ff =
~ ~~~~'~
(16)
~
~
where 3(a ) denotes the imaginary part of the complex quantity a.
If, on the other hand, all ellipsoids are randomly oriented in the mixture, the effective parameters are scalars~ because the composite is bi-isotropic :
Eeff = So + Z «ee. (17)
, -,,,,.
i~eff ~ f~0 + ~j ~
mm,
(~~~
, ,,, =
n lj 3(a~~,)
~eff,, ~
'' (~9)
3 Q~
These equations show that in case of ECP, the effective parameters achieve anomalously high
values. For the case of aligned ellipsoids, the effect is strongest, but only in the direction normal to the flat sides of the inclusions. For randomly oriented disks, ECP shows up for fields with any vector direction, but the magnitude of ECP is three times smaller.
3. Discussion.
The Enhanced Chiral Polarization means a strong change on the optical or electromagnetic properties of mixtures compared with the homogeneous phase. Due to dispersion of the
material parameters, it is easy to imagine that the mixing affects greatly the spectral behavior of media. In this sense, ECP may have similarities to the peculiar effective properties of metal-
insulator mixtures, like percolation [9].
3. I DIELECTRIC ANOMALY. It is known that small metallic-panicle systems exhibit a strong characteristic absorption peak which is absent in the bulk metal [10]. This may change the appearance of gold into ruby-red in suspensions of colloidal particles. With Maxwell-Gamett model, the absorption peak has been explained in granular composites and cermet films. In a
more complete model for light scattering by spherical particles, the absorption peak can be
explained with surface modes inside the sphere ill].
Maxwell-Gamett is able to predict the dielectric anomaly but it is not able to explain percolation [12]. The criticizing voices often label the dielectric anomaly as « Lorentz (') catastrophe » [14]. However, the rivaling theories, like the Bruggeman effective medium model, are not able to show any dielectric anomaly.
Looking at the anomaly in the frequency domain, the strong absorption point is often called
« Fr6hlich frequency » ill ]. However, the ECP is now different from the Frbhlich phenome-
non in metals and metal mixtures due to the following fact. The condition for Fr6hlich
frequency means that the real part of the permittivity of the metal component has to be equal to 2 e~, which happens normally at frequencies much higher than the microwave region. On the other hand, microwaves and millimeter waves form the very region for the applications of chiral materials these days. Therefore, the classical dielectric anomaly is a qualitatively
different phenomenon than ECP, which can appear for ordinary positive values for the
permittivity e and permeability ~ of the material. Now, only the chirality parameter K needs to
obey the limiting condition given above
~
,i
=
fi. (20)
3.2 PRACTICAL LIMtTATIONS. Practical aspects may limit the ideality of the theoretical
model for ECP described in this paper. The infinitely strong polarizability peak only appears for disk-shaped inclusions, that have materials parameters obeying the condition (20). Given this condition for the real parts of the parameter values, it is the magnitude of the imaginary parts of s, ~, K that determine the amplitude of ECP. Due to the dispersive character of
material parameters of chiral media, the imaginary parts cannot be neglected.
The magnitude of ECP depends also on how well the shape of the chiral disks can be considered disk-like. And perhaps the most difficult practical limitation is the condition (20).
This condition has not yet been demonstrated for artificial chiral media, but recent work on novel composites show already values that approach this limit ill-
Also the low-frequency assumption made in the ECP model above limits its range of
applicability. In polarizability modeling, the size of the inclusion needs the small compared
with the wavelength. (A one-fifth is an often-used limit.) Of course, for larger scatterers, the
more complete model along the lines of surface modes ii I] has to be developed for chiral inclusions.
3.3 NONRECIPROCAL coMPosITEs. In passing, it is interesting to note that the enhanced
polarization could be generated also using another class of novel material. The most general bi-
isotropic medium obeys the following constitutive relations [3]
fi
=
Et + (x jK fi fl (21)
fi
=
mfl + (x + jK fit. (22)
(') It may be more appropriate to term this phenomenon «Lorenz» catastrophe rather than
« Lorentz » catastrophe, due to the fact that the temporal order of the discoveries of the Lorenz-Lorentz
polarizability expression is in favor of L. V, Lorenz over H. A. Lorentz II 3].
2606 JOURNAL DE PHYSIQUE III N° 12
Here, a fourth material parameter, the nonreciprocity parameter x appears. X is a measure of the « magnetoelectric el'fect », that has been studied (in the anisotropic case) extensively in
antiferromagnetic crystals [15]. Sometimes the material with X #0 is called «Tellegen
material (?) ». Calculating the polarizability of a disk with the four material parameters, it can
be seen that the expression becomes infinite in the case
X?+K?= ~~ (23)
No so
from which it can be seen that the large polarization can be achieved not only with chirality but also with the nonreciprocity parameter X. However, since the manufacturing of nonreciprocal
composites at microwave frequencies has not reached the maturity of chiral material design, it is probable that we have to wait for « ENP
» some time after the experimental discovery of ECP. When this happens, it remains to be seen.
Acknowledgments.
This work was supported by the Academy of Finland.
References
[Ii N. Engheta Guest Editor, Special Issue on Wave interactions with chiral and complex media, J.
Electromagn. Waves Applications 6, Nos 5/6 (1992)
A. Sihvola Ed., Proceedings of Bi-isotropics'93 : Workshop on novel microwave materials (Espoo, Finland, 1-4 February 1993) ; Helsinki University of Technology, Electromagnetics Laboratory Report Series, Report137 (February 1993)
A. Sihvola, S. Tretyakov and I. Semchenko Eds., Proceedings of Bianisotropics'93 Seminar on electrodynamics of chiral and bianisotropic media (Gomel, Belarus, 12-14 October 1993) Helsinki University of Technology, Electromagnetics Laboratory Report Series, Report159 (December 1993)
F. Mariotte and J.-P. Pameix Eds., Proceedings of Chiral'94 : 3rd Intemational Workshop on chiral, hi-isotropic, and bianisotropic media (Pdrigueux, France, 18-20 May 1994).
[2] 1. Hargittai and C. A. Pickover Eds., Spiral Symmetry (World Scientific, Singapore, 1992).
[3] Lindell I. V., Sihvola A. H., Tretyakov S. A. and Viitanen A. J., Electromagnetic waves in chiral and hi-isotropic media (Artech House, Boston and London, 1994).
[4] Lakhtakia A., Varadan V. K. and Varadan V. V., On the Maxwell-Garnett model of chiral
composites, J. Mater. Res. 8 (1993) 917-922.
[5] Sihvola A. H. and Lindell I. V., Analysis on chiral mixtures, J. Elecn.omagn. Waves Applications 6
(1992) 553-572.
[6] Sihvola A. H. and Lindell I. V., Polarisability and mixing formula for chiral ellipsoids. Electron.
Lent. 26 (1990) 1007-1009
Lakhtakia A., Polarizability dyadics of small chiral ellipsoids, Chem. Phys. Lent. 174 (1990) 583- 586.
[7] Kellogg O. D., Foundations of potential theory, Chapter VII (Dover Publications, New York, 1953)
Landau L. D. and Lifshitz E. M., Electrodynamics of continuous media, Section 4, Second Edition (Pergamon Press, 1984).
(2) Although bianisotropic nonreciprocity has been well observed and documented in the literature, we do not yet have samples of man-made isotropic Tellegen material. As a matter of fact, claims are being
made [16] that Tellegen media cannot exist. This controversial issue has not yet been settled.
[8] Sihvola A. H. and Lindell I. V., Chiral Maxwell-Gamett mixing formula, Electron. Lent. 26 (1990) 118-119
Lakhtakia A., Varadan V. K. and Varadan V. V., Dilute random distribution of small chiral spheres, Appl. Opt. 29 (1990) 3627-3632.
[9] Hammersley J. M., Origins of percolation theory, in Percolation structures and processes, G. Deutscher, R. Zallen, J. Adler, Eds., A?in. Israel Phys. Soc. 5 (1983) 47-57
Stauffer D., Introduction to percolation theory (London and Philadelphia, Taylor & Francis, 1985) Grimmett G., Percolation (New York, Springer Verlag, 1989).
[10] Cohen R. W., Cody G. D., Coutts M. D. and Abeles B., Optical properties of granular silver and gold films, Phys. Rev. B 8 (1973) 3689-3701.
III] Bohren C. F. and Huffman D. R., Absorption and scattering of light by small particles (New York, Wiley, 1983) p. 325.
[12] Sheng P., Theory for the dielectric function of granular composite media, Phys. Rev. Lent. 45 (1980) 60-63.
[13] Sihvola A., Lorenz-Lorentz or Lorentz-Lorenz ? IEEE Antennas Propag. Mag. 33 (1991) 56.
[14] Doyle W. T. and Jacobs I. S., Effective cluster model of dielectric enhancement in metal-insulator
composites, Phys. Rev. B 42 (1990) 9319-9327.
[15] Freeman A. J. and Schmid H., Magnetoelectric Interaction Phenomena in Crystals (London, Gordon and Breach, 1975).
[16] Lakhtakia A. and Weiglhofer W., Are linear, nonreciprocal, biisotropic media forbidden ? IEEE Trans. Microwave Theory Tech. 42, No. 9 (1994)
Sihvola A., Are nonreciprocal bi-isotropic media forbidden indeed? IEEE Trans. Microwave Theory Tech., submitted.
