conduction current density. It is given by equation (4.1.9). The free electrons confined within a neutral conductor form the plasma medium. The artificial dielectrics, including the wire‐medium (i.e. rodded medium), are also modeled as the plasma medium. The ionosphere also supports the plasma medium [B.2]. The plasma medium is modeled using the Drude model, discussed in subsection (6.5.2) of chapter 6. The plasma medium provides a means to engineer materials with negative permittivity. These materials are called the epsilon negative materials, i.e. the ENG materials.
4.2 Electrical Property of Medium
Any material can be electrically characterized by the relative permittivity (εr), relative permeability (μr) and conductivity (σ), or resistivity (ρ). However, these electrical parameters are not constants for any given material. For instance, these are both temperature and frequency‐dependent. Also, these may not be uniform throughout the volume of a material. Further, the characterizing parameters may depend on the field intensity, the direction of the applied field, operating frequency, working temperature, and pressure. The parameters can also depend on the history of a medium. However, the static value of these parameters, at room temperature, is treated as constant. A special kind of material, called chiral material, requires another parameter called chirality, i.e. the handedness of materials for its characterization [B.13, B.17]. Several properties of the medium are described briefly in this section.
4.2.1 Linear and Nonlinear Medium
The relative permittivity (εr) and the relative permeability (μr) of a linear material do not depend on the magnitude of the electric or magnetic field intensity, respectively. However, for nonlinear materials, the relative permittivity and relative permeability are functions of the electric and magnetic field intensity, respectively, and expressed as εr(E) and μr(H). Thus, for an isotropic nonlinear medium, equation (4.1.7a) is written as
where εr(E) = εr1 + εr2E + εr3E2 + ⋯ and so forth. The coefficients εr2, εr3, … indicate the order of nonlinearity in the nonlinear relative permittivity. In the case of a time‐harmonic electric field, i.e. E = E0ejωt, equation (4.2.1) is written as,
(4.2.2)
It is obvious that while the input signal has only one frequency ω, shown in Fig. (4.2), the output of a nonlinear medium has several harmonically related frequency components, ω, 2ω, 3ω, … and so forth. Thus, a sinusoidal input signal gets distorted, once it passes through a nonlinear medium. Such distortion also occurs in an amplifier in the nonlinear region.
Similarly, the relative permeability of a nonlinear magnetic medium is a function of the amplitude of the magnetic field. The constitutive relation, given by equation (4.1.7b), is written as
Figure 4.2 Response of nonlinear medium showing generation of harmonics.
Figure 4.3 Inhomogeneous medium showing a step variation of relative permittivity with substrate height.
where μr(H) = μr1 + μr2H + μr3H2 + ⋯ and so forth. The coefficients μr2, μr3, … indicate the order of nonlinearity in the nonlinear relative permeability of a magnetic medium.
4.2.2 Homogeneous and Nonhomogeneous Medium
The relative permittivity (εr) and the permeability (μr) are not necessarily uniform throughout the volume of a medium. These parameters could also be position‐dependent. The variation in εr and μr could be in discrete steps, or they could be continuous functions of the position. Likewise, the conductivity of a medium can also be a function of position. If the parameters εr, μr, and σ are uniform throughout the medium, the medium is called homogeneous; otherwise, it is a nonhomogeneous medium or an inhomogeneous medium. A multilayer dielectric medium, forming a parallel capacitor, as shown in Fig. (4.3), is a nonhomogeneous medium, where the relative permittivity εr(x) is a function of position x in discrete steps. The conductivity of a doped Si substrate is a function of the depth of penetration of the charged carrier, forming a continuously variable nonhomogeneous medium.
4.2.3 Isotropic and Anisotropic Medium
Inside the isotropic dielectric medium, the electric displacement vector
However, there are dielectrics, such as quartz, sapphire, alumina, MgO, and so forth, where
