Metamaterials

Metamaterials are materials artificially synthesized with unusual dielectric and magnetic properties, generally attained by including metals or usual dielectrics inside a host material.
The metamaterials idea rises from the observation that the concept of homogeneity is absolutely relative, indeed, in a sense, every material is a composite, even if the individual ingredients consist of atoms and molecules. The original objective in defining a permittivity, ε, and permeability, μ, was to present an homogeneous view of the electromagnetic properties of a medium. Therefore it is only a small step to replace the atom of the original concept with structure on a larger scale [1].
If we consider a periodic structure defined by a unit cell of dimension a (Fig.1b), the content of the unit cell will determine the effective response of the system as a whole if the applied field frequency satisfies the condition:

matematerials1

Indeed, in this case, the external field is too myopic to detect internal structure, and, in this limit, the medium can be characterized by means of an effective permittivity and permeability defined as:

formula_1  (1)

Where formula_2 are, respectively, the average of the electric displacement and the magnetic induction vector over the medium unit cell, while eav and hav are the average of the applied electric and magnetic field.

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So, an accurate choice of the geometry, position and parameters of the materials which compose the medium unit cell, allows us to synthesise unusual value of eff ed eff , this can be achieved thanks to the nanotechnologies.

Double Negative Materials (DNG)

DNG is a new class of metamaterials, which have negative values of both the dielectric permittivity and the magnetic permeability, while a material which have only one of these parameters less than zero is referred as Single Negative

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They were introduced in 1968, by Veselago [2] who predicted their unusual properties and named it Left-Handed Media (LHM), since the electric and magnetic fields form a left-handed system with the wave vector (backward wave [3]). So, in a DNG medium, the phase velocity (which is directed as the wave vector k) is antiparallel to the group velocity (which is directed as the poynting vector S).
The index of refraction of a DNG medium has been shown to be negative and the interaction of an electromagnetic (EM) field with such material allows to observe interesting phenomena such as: reversed refraction (Fig.2a), reversed Doppler effect, backward Cherenkov radiation, near-field focusing from homogeneus slabs (Fig.2b).

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Fig. 2: a) Reversed refraction to the interface among an Rigth Handed Medium (RHM)
and a Left Handed Medium (LHM). b) Near-field focusing from homogeneous slabs.

Since a DNG material doesn’t exist, the Veselago’s theoretical disquisitions were not enough to capture the attention of the scientific world, indeed at the time although they were able to synthesize a ENG material as array of thin wire on a dielectric substratum, they didn’t know a particle with strong magnetic properties such to realize an MNG medium.
They were brought to the attention of the scientific community only in 2000 by Smith [4] who synthesized a LH material as a medium composed of two structures separately having ε<0 and μ<0 for the microwave regime (Fig. 3).
In the last years numerous applications for these materials have been predicted or realized: phase-shifters [6], compact-cavity resonators [7], coupled-line couplers [8], leaky-wave antennas [9].

FDTD tool
We have elected an FDTD approach for the analysis of the DNG medium behaviour.
The FDTD method proposed has two basic appealing features, making it amenable for the characterization of metamaterial device:

1_The possibility to introduce three different source in the FDTD lattice [9]:
- Pointwise E ed H hard sources,
- Field radiated by an electric dipole
- Plane wave in a Total Field/Scattered Field formulation

Then we are able to evaluate the response of an object with any shape and dimension to a propagating wave, or to evanescent wave.

2 _Implementation of the ‘lossy Drude model’ to deal with the DNG dispersive behaviour (metamaterials are generally dispersive materials, i.e. their properties are frequency dependent).

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Fig. 3: a) Split Ring Resonator geometry. b) MNG medium synthesised as SRR arrays . c) ENG medium synthesised as thin wire arrays d) DNG medium as composite material. The medium acts as DNG for a plane wave x-directed and with the electric field y-polarised. e), f) DNG medium’s magnetic permeability and electric permittivity, respectively, of. Fig.a,b,c,d taken to [4].
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Fig.4: The figure shows the Electric field’s time history of an m-n-m pulse (sinusoidal signal with a time duration equal to (2m+n) periods) in two point inside a DNG medium: point 1 and point 2 (the point 1 spatially preceding the point 2), it is evident a phase delay in point 1 with respect to the point 2 demonstrating a negative phase velocity . The insert on the top shows the m-n-m pulse spectrum

The tool’s affidability is demonstrated by the results achieved, in agreement with those reported by authoritative studious that for a long time deal him with the matter.
Fig. 4 shows the propagation of an m-n-m pulse (a signal with a small bandwidth) in a matched to free space DNG medium; it is evident that the medium phase velocity is negative.
While, Fig. 5 shows the reflected field of a ENG, MNG e DNG slab (the input signal is a single pulse).
Now we are studying the propagation of a modulated signal in a DNG medium, Fig.5 shows the results achieved for a plane wave x-directed electric field z-polarized (carrier frequency equal to 10 GHz) amplitude modulated by a Gaussian pulse. It is evident the dispersive medium behaviour, indeed the signal experiences a strong distorsion as it propagate in the DNG medium.

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Fig. 5: a) Reflected electric field from a MNG, ENG and DNG slab (in this case, being the slab matched to the free space, the reflected field is negligible), the input signal is a single-pulse (on the left). b) Comparison between the frequency spectrum of the Single pulse assumed as input signal and the index of refraction of the simulated DNG medium.
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Fig. 6: Gaussian pulse propagating in a DNG medium macthed to the free space at 10 GHz. The Electric field’s time history is showed in three points: point 1 preceding the slab’s front face, and in points 2 and 3 located inside the slab. The signal’s time duration grows, as it propagates in the medium, this is a consequence of the dispersive behaviour of the DNG medium

Recommended readings
[1] J. B. Pendry,A.J. Holden, D. J. Robbins, Stewart, W.J.; “Magnetism from conductors and enhanced nonlinear phenomena”, Microwave Theory and Techniques, IEEE Transactions on Volume 47, Issue 11, pp: 2075 – 2084 Nov. 1999.
[2] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Soviet Physics USPEKHI., vol. 10, no. 4, pp. 509-514, Jan-Feb. 1968.
[3] S. Ramo, J.R. Whinnery, T. Van Duzer, “Fields and waves in communication electronics”, John Wiley, 1995.
[4] D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser and S. Schultz, “Composite medium with simultaneously negative permeability and Permittivity’, Phisical Rev. Lett., vol. 84, no. 18, pp. 4184-4187, May 2004.
[5] E. Ozbay, K. Aydin, E.Cubukcu, M. Bayindir, “Transmission and reflection properties of composite Double Negative metamaterials in free space”, “, IEEE Trans. on Anten. and Prop., vol. 51, no. 10, pp. 2592-2595, Oct. 2003.
[6] M. Antoniades and G. V. Eleftheriades, “Compact, Linear, Lead/Lag Metamaterial Phase Shifters for Broadband Applications”, IEEE Anten. and Wireless Prop. Lett., vol. 2, pp. 103-106, July 2003.
[7] N. Engheta, “An idea for thin Subwavelength Cavity Resonators using metamaterials with negative permittivity and permeability”, IEEE Anten. and Wireless Prop. Lett., vol. 1, no. 1, pp. 1494-1504, Feb. 2002.
[8] R. Islam and G. V. Eleftheriades, “A planar metamaterial co-directional coupler that couples power backwards”, 2003 IEEE Int. Microwave Symp. Dig., pp. 321-324, June 2003.
[9] C. Allen, C. Caloz, and T. Itoh, “Leaky-waves in a metamaterial-based two-dimensional structure for a conical beam antenna application”, 2004 IEEE MTT-S Int. Microwave Symp. Dig., vol. 1, pp 305-308, June 2004
[10] A. Taflove, “Computational Electrodynamics, The Finite Difference Time-Domain Method”, Artech House, Norwood, MA, 1995.