href="#fb3_img_img_ae2bd8ab-d08e-52e1-b828-42831298c5b9.png" alt="images"/> fields applied to such materials create both magnetization and electric polarization. The constitutive relation relating four flux and field vectors for a linear magneto‐electric medium is expressed as follows [B.21–B.23]:
The above given four material parameter tensors
(4.2.15)
In the absence of cross‐coupling, i.e. for
The bi‐isotropic materials are isotropic materials also showing cross‐coupling of electric and magnetic fields. However, Fig. (4.5b) shows that the general bi‐isotropic medium has special forms – isotropic, Pasture, Tellegen, and bi‐isotropic. For general bi‐isotropic medium, the medium tensors are reduced to scalars, and the constitutive relations given by equation (4.2.14) are reduced to the following simpler form [B.23]:
(4.2.16)
where ξ2/με is nearly unity. The magnetoelectric coupling parameters ξ and ζ have two components: the chirality parameter κ (kappa) and the cross‐coupling parameter χ (chi). The chirality parameter κ measures the degree of the handedness of the medium. The parameter χ is due to the cross‐coupling of fields. It decides the reciprocity (χ = 0) and nonreciprocity (χ ≠ 0) of the medium, giving the reciprocal and nonreciprocal material medium, respectively. In absence of cross‐coupling, i.e. χ = 0, the parameters ξ and ζ are reduced to imaginary quantities, and the bi‐isotropic medium is reduced to a nonchiral simple isotropic medium for κ = 0 and to a chiral medium for κ ≠ 0. It is also known as Pasteur medium. It supports the left‐hand and right‐hand circularly polarized waves as the normal modes of propagation. It is a reciprocal medium. For κ = 0, χ ≠ 0 another medium, called Tellengen medium, is obtained. It is a nonreciprocal medium. The general bi‐isotropic medium has χ ≠ 0, κ ≠ 0. It is a nonreciprocal medium.
The gyrotropic medium and bianisotropic medium support left‐hand and right‐hand circularly polarized EM‐waves. However, there is a difference. The gyrotropic medium supports the Faraday rotation, i.e. rotation of linearly polarized wave while propagating in the medium, whereas bianisotropic medium does not support it [B.21].
4.2.4 Nondispersive and Dispersive Medium
The medium in which the EM‐wave propagates with equal phase velocity at all frequencies is called the nondispersive medium. For such a medium, the relative permittivity and relative permeability are real quantities, and they are independent of frequency. However, practically every material medium has losses; and thus their relative permittivity and relative permeability are complex quantities, and these are a function of frequency. Thus, normally a material medium supports frequency‐dependent phase velocity. Such a medium is called the dispersive medium. This kind of dispersion is known as material dispersion [B.2, B.3, B.9, B.10]. The material dispersion in the microwave and the mm‐wave ranges is normally negligible for the substrates used in the planar technology. However, a composite substrate, made of the layered dielectric media, is dispersive. Likewise, the artificial metamaterial substrate is also dispersive. The plasma and conducting media are also dispersive in the microwave range. The material dispersion is an important phenomenon near the optical frequency. The constitutive relation for the dispersive medium is
Figure 4.5 Classification of bianisotropic and bi‐isotropic materials.
(4.2.17)
Lorentz oscillator model of a dielectric material, discussed in section (6.5) of chapter 6, helps to understand the frequency‐dependent origin of the ε(ω). When an EM‐pulse, like a Gaussian pulse, passes through a dispersive medium, its pulse‐width widens due to the separation of different frequency components, as each frequency component travels at a different velocity in the dispersive medium. This is known as the pulse‐spreading phenomenon. It limits the speed of digital data transmission through the dispersive medium, as the digital bits can overlap each other. However, a dispersive medium can be nonlinear also, where the pulse spreading can be balanced by the nonlinearity. Under such combined dispersion, and nonlinearity, a pulse can propagate without changing its shape. Such robust EM‐waves are called the solitons. The solitons are useful in optical communication. However, the solitons have also been generated in the microwaves frequency range
4.2.5 Non‐lossy and Lossy Medium
All physical dielectric and magnetic material media have some amount of loss. Normally, the substrates used in the microwave planar technology do not have a high loss, except the doped semiconducting substrates. The lossy dielectrics are known as the imperfect dielectrics. The losses in the dielectric and magnetic materials change the relative permittivity and permeability into complex quantities:
The imaginary parts of the relative permittivity and permeability are taken as negative quantities due to our choice of time‐harmonic function ejωt. However,
