Effective Constitutive Characteristics of Racemic Mixtureof Thin-Wire Helices
Masamitsu Asai 1, Hideaki Wakabayashi 2, Keiji Matsumoto3, Jiro Yamakita2
1

1

Faculty of Biology-Oriented Science and Technology, Kinki University, Japan,2Okayama Prefecture University, Japan,
3
Osaka Sangyo University, Japan

Introduction

The artificial media have been of great interest in thecommunities of electromagnetic wave engineering including thefields of microwaves, milli-meter waves, optics etc. These are
the fabricated aggregation structures composed of artificial
molecules i.e. small objects which are designed to showrequired macroscopic electromagnetic characteristics. The sizes
and spacings between the objects are usually designed to bemuch smaller than the wavelength. Such artificial structures wasstudied first by K. F. Lindman in 1914 through 1922 [1] for theresearch on the dispersion characteristics of the optical activityof the artificial chiral media as the magnified scale models of
the organic compounds. Research on artificial chiral media as
well as that on artificial dielectrics and magnetic materials has
been on-going through the golden years of the chiral media in‘90s [2], [3]. The greatest concern on the artificial media has
moved to the left-hand materials (LHM) with the term
‘metamaterials’ since J. B. Pendry et al. theoretically treated
LHM [4], and D. R. Smith et al. demonstrated the negativeindex of refraction in a fabricated LHM in microwave regions at
the end of the ‘90s [5],[6]. On the other hand, chiral media havebeen of great interest in the researchers of carbon nano coils(CNCs) and micro coils (CMCs) especially since the synthesisof CMCs in high yield content was discovered and the fineabsorption characteristics at frequencies of tens throughhundreds GHz was shown [7],[8]. The effective constitutive
parameters of isotropic distributions of chiral mixtures of thinwire helices have been analyzed by the quasi-static Lorentz’approach under the conditions that helices with an identicaldimension cluster in groups [9], and that helices in clusters haveelastic deviations [10],[11] as the models of isotropic chiralmixture of carbon coils. The bulks of CMCs synthesized for alarge amount however are usually composed of almost the sameamount of right- and left-handed coils i.e. racemic mixture andare macroscopically achiral. The method to separate the rightand left-handed coils has been explored and not yet discovered.
Recent development of material sciences has made it possible torealize the array of clusters of carbon coils with identicalorientation, and such array structures would bear the possibilityof new bianisotropic media [2]. It is reported that the multipole
moments to the order of electric quadrupole and magnetic dipole
are necessary to explain the anisotropic chirality in natural
materials [12],[13]. In most of the analyses for bianisotropic
cases however, the structures of inclusions do not require theorder of multipole moments higher than that of dipoles [14],[15], or the contributions of multipole moments themselves are
not shown for different structural parameters [16], [17]. Chiralmedia composed of single-helix particles or particles of pairs ofhelices were analyzed and the contributions of quadrupole
moments in the constitutive characteristics were investigated[18]
In this work, the effective constitutive characteristics areanalyzed for the isotropic and anisotropic racemic mixture ofmetallic thin-wire helices. Inclusion particles are composed of
one or two metallic thin-wire helices in free space. The sizes andspacings between helices are assumed to be small compared to

the wavelength. The quasi-static Lorentz’ approach with fineevaluation of interaction between particles [19] is combinedwith the method of moments with thin-wire approximation to
calculate the effective parameters. The contributions of theelectric quadrupole moments in the determination of theeffective constitutive parameters are also investigated.

2Analysis

Both isotropic and anisotropic distributions of particles areassumed. For the anisotropic cases, three-dimensional periodic
array with the periodicities along the Cartesian coordinates x, yand z being dx , dy and dz respectively is assumed. The sizes andthe spacings of the particles are assumed to be much smaller
than the wavelength for the quasi-static approximation. The
permittivity and the permeability for free space and timedependence are assumed to be ( ε 0, μ 0) and exp(j ω t)respectively. Each particle is assumed to be composed of aright-handed metallic thin-wire single helix or a skew pair ofFigure 1: A pair of parallel helices
right and left-handed metallic thin-wire helices with number ofturns T , height L , radius A , pitch P , wire radius W , andwire length l . The axis of the single helix is oriented in z
direction. As for the pairs of helices, parallel pairs are also
assumed as the special case of skew ones. Figure 1 shows theparallel pair of helices where the axes of two helices withdistance D are parallel. The end of the wire of a single-helix
particle or the right helix of a pair of helices (Fig. 1) in thez < 0 region are assumed to be located in the position( A cos ϕ , A sin ϕ , - L 2) where the angle ϕ in the plane parallel
to xy plane is defined in the clockwise direction on the z axis
from the xz plane. The skew cases can be realized by rotatingthe right helix clockwise on the x axis for angle θ . The quasistatic Lorentz’ approach with fine evaluation of interaction
between particles [19] is combined with the method of moments
to calculate the effective parameters. In the method of moments
only the axial components of currents on the wire surfaces are
considered and the circumferential variations are neglected forthe thin wire approximation.

3

Numerical Investigations

Calculations are performed under the conditions thatl =25.0(mm), L =4.0(mm), P =L/T and W =0.07(mm) , where
the periodicities of the three-dimensional array are chosen as
dx=dy=4.0A, dz=2.0L for the single-helix cases, and dx=dy=2.5U
dz=2.5L for the cases of pairs of helices. Results will be shownin the full manuscript and in the presentation.

4Conclusion

The effective constitutive characteristics are analyzed for theisotropic and anisotropic racemic mixture of metallic thin-wire
helices. Inclusion particles are composed of one or two metallic
thin-wire helices in free space. The sizes and spacings between
helices are assumed to be small compared to the wavelength.
The quasi-static Lorentz’ approach with fine evaluation ofinteraction between particles is combined with the method ofmoments with thin-wire approximation to calculate the effectiveparameters. The contributions of the electric quadrupole
moments in the determination of the effective constitutive
parameters are also investigated.

References[1] I. V. Lindell, A. H. Sihvola, S. A. Tretyakov and A. J. Viitanen,"Electromagnetic Waves in Chiral and Bi-Isotropic Media," Artech
House, pp. 1-18, 193-243, 1994.
[2] H. Cory, “Chiral devices – an overview of canonical problems,” J.
Electro. Waves Applic., Vol. 9, No. 5/6, pp. 805-829, 1995.
[3] M. Asai, J. Yamakita, K. Matsumoto and H. Wakabayashi, "Onelectromagnetic chirality of helix-loaded materials," MaterialsIntegration, Vol. 17, No. 7, pp. 27-33, 2004.
[4] J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev.
Lett., Vol. 85, No. 18, pp. 3966-3969, 2000.
[5] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser and S.
Schultz, “Composite medium with simultaneously negativepermeability and permittivity,” Phys. Rev. Lett., Vol. 84, No. 18, pp.
4184-4187, 2000.
[6] L. W. Li, H. Y. Yao, Q. Wu and Z. N. Chen, “Broad-bandwidth andlow-loss metamaterials: Theory, design and realization,” J. Zhejiang
Univ. SCIENCE A, Vol. 7, No. 1, pp. 5-23 ,2006.
[7] S. Motojima, M. Kawaguchi, K. Nozaki and H. Iwanaga, Appl. Phys.
Lett., Vol. 56, No. 4, pp.321 -, 1995.
[8] S. Motojima, Papers of Technical Meeting on E.M.T., Vol. EMT-0383, pp.65-,2003.
[9] M. Asai and J. Yamakita, “Numerical study on waves from particles
composed of bunches of helices,” Proc. PIERS2006, Vol. 1, 2006.
[10] M. Asai and J. Yamakita, “Characteristics of effective media
structured by clustering helices with elasticity,” Technical Reportof the Radiation Science Society of Japan RS07-06, 2007.
[11] M. Asai and J. Yamakita, “Effective medium parameters for
distributions of clustering helices with structural deviations,” Proc.
ISAP2007, Vol. 1, 2007.
[12] A. D. Buckingham and M. B. Dunn, “Optical activity of orientedmolecules,” J. Chem. Soc. A, pp. 1988-1991, 1971.
[13] E. B. Graham and R. E. Raab, “Covariant D and H fields forreflection from a magnetic anisotropic chiral medium,” J. Opt. Soc.
Am. A, Vol. 14, No. 1, pp. 131-134, 1997.
[14] S. A. Tretyakov and F. Mariotte, “Maxwell Garnett modelling ofuniaxial chiral composites with bianisotropic inclusions,” J. Electro.
Waves. Applic., Vol. 9, No. 7/8, pp. 1011-1025, 1995.

[15] C. Y. Chung and K. W. Whites, “Effective constitutive parameters
for an artificial uniaxial bianisotropic chiral medium,” J. Electro.
Waves. Applic., Vol. 10, pp. 1363-1388, 1996.
[16] I. P. Theron and J. H. Cloete, “The optical activity of an artificial
non-magnetic uniaxial chiral crystal at microwave frequencies,” J.
Electro. Waves. Applic., Vol. 10, No. 4, pp. 539-561, 1996.
[17] J. Reinert and A. F. Jacob, “Multipolarizability tensors of thin-wirescatterers: A direct calculation approach,” IEEE Trans. on
Antennas. Propag., Vol. 49, No. 11, pp. 1532-1538, 2001.
[18] M. Asai, H. Wakabayashi, K. Matsumoto and J. Yamakita,“Contributions of the quadrupole moments in the effective
constitutive characteristics for the anisotropic distributions ofmetallic thin-wire helices,” Proc. Int. Conf. Modeling andSimulations Technology, Vol. OS5, pp. 179-182, 2011[19] A. Ishimaru, S. W. Lee, Y. Kuga and V. Jandhyala, “Generalizedconstitutive relations for metamaterials based on the quasi-staticLorentz Theory,” IEEE Trans. on Antennas. Propag., Vol. 51, No.
10 pp. 2550-2557, 2003.