**2.2 Metamaterial's classification based on their properties**

Russian scientist Veselago first proposed the metamaterial classification by considering the permittivity, ε, and the permeability, μ of a homogeneous material.

**Figure 3.** *Metamaterial classification.*

The relationship between the refractive index and the constituent parameters ε and μ is given by the formula:

$$m = \pm \sqrt{\varepsilon\_r \mu\_r} \tag{4}$$

where ε<sup>r</sup> and μ<sup>r</sup> are the relative permittivity and permeability of the material. From Eq. (4), sign � of n can get 1 in the four cases, which depends on the pairs of the sign of ε<sup>r</sup> and μr. The electromagnetic metamaterials are classified based on each case of the pair sign ε and μ; they are shown in **Figure 3**.

In quadrant I, both parameters ε and μ are positive and are called double positive (DPS) or right-handed medium (RHM). In quadrant II, the parameters are ε < 0 negative, and μ > 0—positive, and such material is called epsilon negative (ENG) medium and is represented by plasma. In quadrant III, parameters ε < 0—negative, and μ < 0—negative, this region is called double-negative (DNG) or left-handed medium (LHM), and such material could not be found in nature. The quadrant IV ε > 0—positive, and μ < 0—negative, such material is called μ—negative (MNG), represented by ferrite materials.

#### **2.3 Types of metamaterial**

A split-ring resonator (SRR) is a type of metamaterial, which is artificially created. SRR cell is made up of a pair of enclosed loops of nonmagnetic metals that split at

*Analysis and Design of Miniaturized Substrate Integrated Waveguide CSRR Bandpass Filters… DOI: http://dx.doi.org/10.5772/intechopen.104733*

#### **Figure 4.**

*Split-ring resonator with its equivalent circuit.*

opposite ends, as shown in **Figure 4**. When these materials are exposed to the magnetic field of electromagnetic waves, they give strong magnetic coupling unavailable in conventional materials. When SRRs are arranged periodically (array), they provides negative permeability.

The above structure of SRR is known as edge-coupled split-ring resonator (EC-SRR) structure, which comprises concentric metal split rings printed on the same side of the dielectric substrate. EC SRR benefits of strong magnetic polarizability near resonance and easy fabrication. However, it has certain drawbacks: (i) Its electric size cannot be reduced below one-tenth of wavelength; (ii) it suffers from cross-polarizability/ bianisotropic effect. Another type of SRR overcomes these limitations, called broadsidecoupled SRR (BC-SRR) [10]. In the BC-SRR configuration, the rings are etched on both faces of the substrate, as shown in **Figure 5a**. Similar to EC-SRR, charges formed in the lower half of the BC-SRR are the replica of charges formed in the upper half, as shown in **Figure 5b**. Though this formation of charge does not create an electric dipole, BC SRR is non-bianisotropic. Since both rings are of identical dimension and keep inverse symmetry, for this reason, cross-polarizability tensor vanishes.

The application of the Babinet principle leads to the origin of its counterpart known as a complementary split-ring resonator (CSRR) in which the rings are

engraved on the conductive surface, and its magnetic and electric characteristics are changed when compared with SRR.
