We analyze the reflection and transmission coefficients calculated from transfer matrix simulations on finite lengths of electromagnetic metamaterials, to determine the effective permittivity (ε) and permeability (µ). We perform this analysis on structures composed of periodic arrangements of wires, split ring resonators (SRRs) and both wires and SRRs. We find the recovered frequency-dependent ε and µ are entirely consistent with analytic expressions predicted by effective medium arguments. Of particular relevance are that a wire medium exhibits a frequency region in which the real part of ε is negative, and SRRs produce a frequency region in which the real part of µ is negative. In the combination structure, at frequencies where both the recovered real parts of ε and µ are simultaneously negative, the real part of the index-of-refraction is found also to be unambiguously negative. * Permanent address: Institute of Physics, Slovak Academy of Sciences, Dúbravska cesta 9, 842 28 Bratislava, Slovakia.It has been proposed that electromagnetic metamaterials-composite structured materials, formed from either periodic or random arrays of scattering elements-should respond to electromagnetic radiation as continuous materials, at least in the long wavelength limit [1,2]. In recent experiments and simulations [3,4], it has been demonstrated that certain metamaterial configurations exhibit scattering behavior consistent with the assumption of approximate frequencydependent forms for ε and µ. However, the techniques applied in those studies probed the materials indirectly, and did not provide an explicit measurement that would assign values for ε and µ. It is our aim in this paper to show that the previous conjectures were indeed valid: unambiguous values for ε and µ can be applied to electromagnetic metamaterials. Our approach here utilizes the transmission and reflection coefficients (Sparameters, equivalently) calculated for a wave normally incident on a finite slab of metamaterial. We invert the scattering data to determine n and z, for systems of several thicknesses, from which we obtain self-consistent values for ε and µ. While we utilize simulation data in this study, the technique we describe will be readily applicable to the experimental characterization of metamaterial samples whenever the scattering parameters are known.

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...