Resonant and antiresonant frequency dependence of the effective parameters of metamaterials

Abstract

We present a numerical study of the electromagnetic response of the metamaterial elements that are used to construct materials with negative refractive index. For an array of split ring resonators (SRR) we find that the resonant behavior of the effective magnetic permeability is accompanied by an anti-resonant behavior of the effective permittivity. In addition, the imaginary parts of the effective permittivity and permeability are opposite in sign. We also observe an identical resonant versus anti-resonant frequency dependence of the effective materials parameters for a periodic array of thin metallic wires with cuts placed periodically along the length of the wire, with roles of the permittivity and permeability reversed from the SRR case. We show in a simple manner that the finite unit cell size is responsible for the anti-resonant behavior.PACS numbers: 41.20.Jb, 42.70.Qs, 73.20.Mf The recent development of metamaterials with negative refractive index -or double-negative (DNG) metamaterials 1 has confirmed that structures can be fabricated that can be interpreted as having both a negative effective permittivity ǫ and a negative effective permeability µ simultaneously. Since the original microwave experiment of Smith et al., 2 various new samples were prepared, 3, 4 all of which have been shown to exhibit a pass band in which the permittivity and permeability are both negative. These materials have been used to demonstrate negative refraction of electromagnetic waves, 5 a phenomenon predicted by Veselago.6 Subsequent experiments 7 have reaffirmed the property of negative refraction, giving strong support to the interpretation that these metamaterials can be correctly described by negative permittivity and negative permeability. 8, 9There is also an increasing amount of numerical work [10][11][12] in which the transmission and reflection of electromagnetic wave is calculated for a finite length of metamaterial. For a finite slab of continuous material, the complex transmission and reflection coefficients are directly related to the refractive index n and impedance z associated with the slab, which can in turn be expressed in terms of permittivity ǫ and permeability µ. A retrieval procedure can then be applied to find material parameters for a finite length of metamaterial, with the assumption that the material can be treated as continuous. A retrieval process was applied in Ref., 13 and confirmed that a medium composed of split ring resonators (SRRs) and wires could indeed be characterized by effective ǫ and µ whose real parts were both negative over a finite frequency band, as was the real part of the refractive index n.The retrieval process, however, uncovers some unexpected effects. For the SRR medium, for instance, the real part of the effective permittivity ǫ ′ exhibits an anti-resonant frequency dependence in the same frequency region where the permeability undergoes its resonance. This anti-resonance can be seen in the composite SRR+wire negative index medium as well. The anti-resonance in the real...

Effective Medium Theory of Left-Handed Materials

“…Using these data a retrieval procedure can then be applied to obtain the effective permittivity ǫ and permeability µ, under the assumption that the metamaterial can be treated as homogeneous. This procedure confirmed [14,15] that a medium composed of SRRs and wires could indeed be characterized by effective ǫ and µ whose real parts were both negative over a finite frequency band, as was the real part of the refractive index n.In the present paper, we study the transmission of periodic systems made up of wires alone, SRRs alone and of combined systems of wires and SRRs (LHMs). Through a very detailed retrieval scheme [14], the effective ǫ and µ of those systems are obtained.…”

Electromagnetically induced left-handedness in a dense gas of three-level atoms

“…It is natural to have transmission losses in our model, similar to other left-handed structures, as they are unavoidable due to the Kramers-Kronig relations ensuring the causality in the system. On the other hand, theoretically it is not a trivial task to estimate the amount of losses [16,17,18] and to rigorously prove the causality in left-handed materials [19,20]. We can give a simple and rough estimate by simply taking into account the imaginary part of the refractive index which gives that after several microns the optical field will be damped by ∼ 33% due to linear absorption.…”

Elastic metamaterials and dynamic homogenization: a review

“…The most common route to determining these parameters is by the use of retrieval methods [55][56][57][58], where the assumption is that local effective properties may be used to define periodic composites such as metamaterials. These properties are determined by analyzing the complex reflection and transmission coefficients of a finite slab of the composite.…”