2.2 Realization of artificial ε-negative (ENG) medium

As discussed in the previous section, the realization of metallic resonant inclusions patterned in a homogenized host medium had made it possible to synthesize magnetic permeability in the microwave and optical regimes [3]. Similarly, it is possible to engineer the permittivity of a bulk medium by facilitating patterned metallic inclusions. In solid metals, negative permittivity response commonly occurs at the visible and near-ultraviolet regime, due to the entire oscillation of plasmons [15]. Pendry et al. showed that an array of thin rods or wires arranged in a cubical lattice can indeed exhibit negative effective permittivity response at the microwave regime given by the Drude function [2]:

$$\varepsilon(o) = 1 - \frac{o\_p^2}{o(o + jI)}\tag{13}$$

where Γ is the energy dissipation factor of the plasmon into the system (i.e., damping factor). In solid metals, like aluminum, the dissipation factor, Γ, is usually small as compared to the plasmon frequency, ωp. If losses were neglected, (i.e., Γ ≈ 0), it is evident from Eq. (13) that electromagnetic waves below the plasma frequency (ω<sup>p</sup> > ωp) cannot propagate, since ε < 0 (μ here >0). It is also evident from Eq. (13) that the refractive index, n, will be imaginary and a wave in such a medium will be evanescent.

The electric permittivity response in Eq. (13) can be used to represent the effective electrical permittivity response of a synthesized homogeneous medium comprising an array of very thin metallic wires. Note that in the realization of the composite periodic lattice of wires, wires' diameter is essentially much smaller than the operating wavelength in order to mimic an effective homogenized negative permittivity media from such periodic metal rods. The term ω<sup>p</sup> represents the plasma frequency for metals and can be expressed in terms of the electron properties by following relation [2]:

$$
\alpha\_p^2 = \frac{nq^2}{\varepsilon\_0 m\_e} \tag{14}
$$

where q is the electron charge, ε<sup>0</sup> is the free-space permittivity, and n and me are the effective density and mass of electrons, respectively.

From classical electromagnetic theory, metallic wires behave collectively as small resonant dipoles when excited with an applied electric field parallel to the wires plane, similar to the electric dipoles response of atomic and molecular systems in natural materials [2]. Although the metallic wire structure, discussed earlier, can tailor effective permittivity response within the radio frequency and microwave regime, the arrangement of metallic wires in a cubic structure is still bulky and may appear undesirable for planar radio frequency and microwave applications.

Recently, Falcone et al. introduced in [16–18] a subwavelength resonant planar particle, known as the complementary split-ring resonator (CSRR) as shown in Figure 4b, which is the dual counterpart of the SRR. In other words, by following Babinet's principle [19], the complementary of the planar SRR structure is obtained by replacing the SRR metallic rings with apertures and the apertures (surrounding free-space region of SRR) with metal plates. By etching the SRR rings from the metallic ground screen, complementary SRRs form the basis of realizing compact microstrip-based bandstop filters [16]. Such bandstop behavior is attributed to the existence of negative electrical permittivity response, due to an axial time-varying electric field parallel to the CSRR ring plane. Interestingly, as it is practically straightforward to excite CSRR particle with an axial electric field, CSRR particle is very easy to integrate with other planar microstrip circuits.

With the assumption that the largest dimension in CSRR unit cell is much smaller than the operating wavelength, a quasi-static equivalent circuit model can be considered to estimate the effective permittivity response of such inclusion [18].

Consider an axial external uniform time-varying electric field that is parallel to the CSRR inclusion plane, as shown in Figure 4b; the composite structure will react and oppose the applied external electric field by creating internal electric dipole moments that give rise to electrical polarization effect. Following the analytical formulation given in [20] and assuming a homogenized artificial CSRR structure, the overall effective electrical permittivity response of the homogenized artificial CSRR structure can then be written in the form below in terms of basic RLC circuit elements: Electromagnetic Field Interaction with Metamaterials DOI: http://dx.doi.org/10.5772/intechopen.84170

Figure 4.

Two-dimensional view of (a) an artificial magnetic material inclusion, the circular SRR; (b) complementary SRR (CSRR), with dimensions; rout is radius of outer ring, rin is internal ring radius, a represents metallic (aperture) width, b is spacing between rings in SRR (spacing between etched rings in CSRR), and g represents the SRR ring's cut (CSRR etched rings' left metallization). The gray area represents structure metallization.

$$\varepsilon\_{\text{eff}} = \mathbf{1} + \frac{K \, Z\_{\text{inc}}}{Z\_{\text{inc}} + \frac{1}{j \alpha \mathbf{C}\_{\text{eff}}}} \tag{15}$$

where K is a normalized fractional surface area of CSRR inclusion and Ceff is the effective capacitance of a parallel plate capacitor with surface area being the CSRR inclusion surface, while the thickness of the capacitor is the periodicity of the CSRR as a composite infinitely large structure. The term Zinc in Eq. (15) represents the effective impedance of the CSRR inclusion and is given by

$$\mathbf{Z\_{inc}} = \mathbf{R\_{eff}} + j\boldsymbol{j}o\,\,\mathbf{L\_{eff}} \tag{16}$$

where Reff is the effective ohmic losses due to the finite conductivity of the metallic rings around the CSRR slots and is given by the alternating current resistance of the rings. The term Leff in Eq. (16) accounts for the mutual inductive effects between the external and internal strips around the slotted rings. More analytical modeling of the effective electric permittivity and CSRR equivalent circuit parameters can be found in [18, 20]. Table 1 summarizes the most common artificial magnetic and electric inclusions (i.e., SRR and its dual, CSRR) in order to realize single-μNG and -εNG media, respectively, and their equivalent basic circuit representations (Figures 5–7).

#### 2.3 Realization of artificial double negative (DNG) medium

In 1968, Veselago investigated theoretically that electromagnetic waves can propagate in a medium, where both permittivity and permeability are negative [4], which was then termed as metamaterials, DNG media, or LHM materials. The realization of artificial DNG media evolved after the theoretical studies by Pendry in [2, 3] in order to synthesize AMMs and artificial ENG media at low frequencies, and afterward practical realizations and demonstrations of DNG media by Smith and his team in [21, 22], which was achieved by producing a large bulk structure composed of repeated patterns of artificial MNG, in the form of SRRs, and ENG media, in the form of planar metallic strips.

Several interesting properties that are not yet in nature can be achieved with DNG media, including but not limited to backward wave propagation and negatively refracted waves and perfect lens [23].
