Session 3A2b
Mathematical and Numerical Tools for Metamaterials
1
Transformation Optics for Cloaking and Hyperlensing with Metamaterials
Charles Croënne, Davy P. Gaillot, Fuli Zhang, Wounjhang Park, Didier Lippens, . . . . . . . . . . . . . . . . .
Determination of the Effective Constitutive Parameters of Bianisotropic Metamaterials from Reflection
and Transmission Coefficients
Zhaofeng Li, Ekmel Ozbay, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Homogenization of Finite Metallic Fibers and 3D-effective Permittivity Tensor
Guy Bouchitte, Cristophe Bourel, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Homogenization of 3D-dielectric Photonic Crystals and Artificial Magnetism
Guy Bouchitte, Cristophe Bourel, Didier Felbacq, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Progress In Electromagnetics Research Symposium Abstracts, Beijing, China, March 23–27, 2009
424
Transformation Optics for Cloaking and Hyperlensing with
Metamaterials
Charles Croënne1 , Davy Gaillot1 , Fuli Zhang1, 2 , Wounjhang Park3 , and Didier Lippens1
1
Institut d’Electronique, de Microélectronique et de Nanotechnologie (IEMN — UMR CNRS 8520)
Université des Sciences et Technologies de Lille
Avenue Poincaré, BP 60069, 59655 Villeneuve d’Ascq, France
2
Department of Applied Physics, Northwestern Polytechnical University
Xi’an 710072, China
3
Department of Electrical & Computer Engineering, University of Colorado
UCB 425, Boulder, CO 80309-0425, USA
Abstract— We report here on the possibilities afforded by the numerical treatment of space
transformations aimed at better controlling the light from microwave to optics. We illustrate
these possibilities with two examples. The first one concerns cloaking applications which consist
to coat a scatterer with a metamaterial. By proper designing the effective parameters of the
cloak, a perfect matching at the interface between the embedding medium and the cloak can be
achieved. Also, the light is deviated with a parietal flow along the cloak and a reconstruction of
traveling waves behind the cloak. The difficulty to realize a perfect cloak, by monitoring both the
permittivity and permeability gradients, is first pointed on the basis of the current metamaterial
technologies. Secondly, we focus our attention about the frequency dependence of the Radar
Cross Section (RCS) which is ultra narrow in essence. It is shown that the RCS depends on
the dispersion but also on the scale of the cloaked scatterer. The second application concerns
hyperlensing based on highly anisotropic metamaterial media. The condition of hyperlensing,
namely the fact that the propagation of light shows a channeling effect is established. We then
address the problem of magnification which permits one to increase the separation between two
point sources with a sub-wavelength spacing. The originality of the work is the proposal of a
flat configuration when the demonstrations, found in the literature, used round-shaped multilayered microstructures. We conclude with the prospect to develop the reverse problem namely
hyperfocusing.
Progress In Electromagnetics Research Symposium Abstracts, Beijing, China, March 23–27, 2009
425
Determination of the Effective Constitutive Parameters of
Bianisotropic Metamaterials from Reflection and Transmission
Coefficients
Zhaofeng Li and Ekmel Ozbay
Nanotechnology Research Center
Department of Physics, and Department of Electrical and Electronics Engineering
Bilkent University, Bilkent, Ankara 06800, Turkey
Abstract— We propose a method to retrieve the effective constitutive parameters of a slab of
bianisotropic metamaterial composed of split ring resonators from the S parameters. Unlike the
previous method [4], we only use the S parameters in one direction, which makes our method
much simple. Analytical inversion equations are derived and firstly verified for a homogeneous
bianisotropic media. Then, we use this method to extract the effective constitutive parameters
of the bianisotropic metamaterials. The retrieved results corroborate well the conclusions in
previous published papers.
REFERENCES
1. Smith, D. R., S. Schultz, P. Markos, and C. M. Soukoulis, Phys. Rev. B, Vol. 65, 195104, 2002.
2. Katsarakis, N., T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, App. Phys.
Lett., Vol. 84, 2943, 2004.
3. Marques, R., F. Medina, and R. Rafii-El-Idrissi, Phys. Rev. B, Vol. 65, 144440, 2002.
4. Chen, X., B.-I. Wu, J. A. Kong, and T. M. Grzegorczyk, Phys. Rev. B, Vol. 71, 46610, 2005.
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Progress In Electromagnetics Research Symposium Abstracts, Beijing, China, March 23–27, 2009
Homogenization of Finite Metallic Fibers and 3D-effective
Permittivity Tensor
Guy Bouchitte and Cristophe Bourel
Universite de Toulon, France
Abstract— We consider a finite domain of R3 (scatter) filled periodically with high conductivity
metallic fibers. Our aim is to describe mathematically the asymptotic of the harmonic diffraction
problem (electromagnetic waves with a prescribed exp(−iωt) dependence) as the period η tends
to zero. In the limit process, the conductivity in the fibers increases to infinity.
In this talk we will present two results, both obtained for a vanishing volume fraction of fibers
(the section of fibers is infinitesimal with respect to the period): in the first situation fibers are
e3 -parallel connected with length L. We find that the vertical component E3 of the limit electric
field induces a volumic current j density (solution of a 1D-propagation equation in which E3
acts as a source term). The resulting diffraction problem involves (E, H) and j. Unfortunately
the effective permittivity law is non local (convolution kernel with interaction distance of order
L). This important feature is hidden when we assume a polarized electric field and L = +∞: in
this case the effective law can be interpreted through a scalar effective permittivity εeff (ω) which
depend explicitely of the wave number. It becomes negative below a cut-off frequency. Notice
however that fully 3D-negative tensors cannot be reached by this procedure (even if fibers are
disposed in three orthogonal directions).
In our second variant we show that effective 3D-isotropic negative permittivity tensors can be
reached: the trick consists in using the previous (non local) homogenized model at a very small
scale by reiterating periodically our previous fibered structure (at macroscopic scale fibers are
then very small and disconnected). We obtain a local effective law characterized by an effective
permittivity tensor εeff (ω) whose eigenvalues depend on frequency and are ruled by a local spectral
problem. The tensor εeff (ω) turns out to be negative wihin some range of frequencies (band gaps).
Case I: The inclusions are vertical parallel metallic fibers of radius r ¿ η (vanishing filling ratio)
but scaled so that the average 2D-capacity of the cross sections remain positive. The permittivity
ε has a large imaginary part. We find that the vertical component E3 of the limit electric field
induces a volumic current j density (solution of a 1D-propagation equation in which E3 acts
as a source term). The resulting diffraction problem involves (E, H) and j. It is non local in
(E, H). However, in the case of a polarized electric field, we recover an homogenized medium
characterized by an effective permittivity εeff (ω). This εeff (ω) depends explicitely of the wave
number. It becomes negative below a cut-off frequency.
Case II: The filing ratio of inclusions is positive (r ∼ η) but now the permittivity ε is scaled
like η12 . Starting with e3 -invariant geometry we find an homogenized medium with a magnetic
activity. It is described by an effective permeability µeff (ω) depending explicitely on a elementary
cell spectral problem. An important consequence is that, for certain ranges of frequencies, µeff (ω)
becomes negative (band-gap structure).
REFERENCES
1. Bouchitte, G. and D. Felbacq, “Homogenization near resonances and artificial magnetism from
dielectrics,” C. R. Math. Acad. Sci. Paris, Vol. 339, No. 5, 377–382, 2004.
2. Bouchitte, G. and D. Felbacq, “Left handed media and homogenization of photonic crystals,”
Optics Letters, Vol. 30, 10, 2005.
3. Bouchitte, G. and D. Felbacq, “Homogenization of wire mesh photonic crystals embdedded in
a medium with a negative permeability,” Phys. Rev. Lett., Vol. 94, 183902, 2005.
4. Bouchitte, G. and D. Felbacq, “Low frequency scattering by a wire mesh photonic crystal:
Homogenized limit in the capacitary case,” submitted to SIAM J. Applied Maths.
5. Bouchitte, G. and D. Felbacq, “Homogenization of a set of parallel fibers,” Waves in Random
Media, Vol. 7, No. 2, 1–12, 1997.
6. Bellieud, M. and G. Bouchitte, “Homogenization of elliptic problems in a fiber reinforced
structure. Nonlocal effects,” Ann. Scuola Norm. Sup. Pisa Cl. Sci., Vol. XXVI, No. 4, 407–
436, 1998.
7. Bouchitte, G. and M. Bellieud, “Homogenization of a soft elastic material reinforced by fibers,”
Asymptotic Analysis, Vol. 32, No. 2, 153–183, 2002.
Progress In Electromagnetics Research Symposium Abstracts, Beijing, China, March 23–27, 2009
427
Homogenization of 3D-dielectric Photonic Crystals and Artificial
Magnetism
G. Bouchitté1 , C. Bourel1 , and Didier Felbacq2
1
Departement of Mathematics, Université de Toulon, BP 20132, 83957 La Garde Cedex, France
2
GES UMR 5650, Place Bataillon, 34095 Montpellier Cedex 05, France
Abstract— In [1–4], a theory for artificial magnetism in two-dimensional photonic crystals
has been developed for large wavelength (homogenization). The main idea was that a periodic
crystal with high permittivity inclusions shows up micro-resonance effects from which an effective
permeability law with anomalous dirpersion could be evidenced in a explicit way. The main
drawback was however that in this model we assumed magnetic parallel polarization so that
merely infinite photonic crystals (invariants in one direction) could be considered.
In this work we propose a full 3D generalization of previous results: the diffraction of a finite
3D- dielectric crystal is considered at a fixed wavelength and a limit analysis as the period tends
to zero is performed. We evidence a new microscopic vector spectral problem which turns out
to rule the macroscopic behavior of the crystal. We obtain then an extension to the 3D-case of
the results in [1, 3] by proving rigorously that permeability tensor laws can be reached where the
effective tensor exhibits negative eigenvalues in appropriate range of frequencies. This suggests
that periodic bulk dielectric inclusions could be an efficient alternative to the very popular metallic
split-ring structure proposed by Pendry [5].
REFERENCES
1. Bouchitté, G. and D. Felbacq, “Homogenization near resonances and artificial magnetism from
dielectrics,” C. R. Math. Acad. Sci. Paris, Vol. 339, No. 5, 377–382, 2004.
2. Felbacq, D. and G. Bouchitté, “Left handed media and homogenization of photonic crystals,”
Optics Letters, Vol. 30, 10, 2005.
3. Felbacq, D. and G. Bouchitté, “Homogenization of wire mesh photonic crystals embdedded in
a medium with a negative permeability,” Phys. Rev. Lett., Vol. 94, 183902, 2005.
4. Felbacq, D. and G. Bouchitté, “Negative refraction in periodic and random photonic crystals,”
New J. Phys., Vol. 7, 159, 10.1088, 2005.
5. OBrien, S. and J. B. Pendry, “Magnetic activity at infrared frequencies in structured metallic
photonic crystals,” J. Phys. Condens. Mat., Vol. 14, 6383–6394, 2002.
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Progress In Electromagnetics Research Symposium Abstracts, Beijing, China, March 23–27, 2009