the remaining two relative permittivity components have identical values εr⊥. The relative permittivity components εr⊥ are in the plane normal (⊥) to the y‐axis, and also normal to the external electric field Ey. The y‐axis is the main axis. It is also known as the C‐axis, or the optic‐axis, or the extraordinary axis. In the direction of the optic axis, the permittivity is different. The other two x‐ and z‐axes are known as the ordinary axes. The x‐ and z‐polarized EM‐waves, known as the ordinary waves, in the (x‐z)‐plane travel with the same velocity, whereas y‐polarized EM‐wave, known as the extraordinary wave, travels with another velocity. The nonhomogeneous medium as shown in Fig. (4.3) is also a uniaxial medium with the x‐axis as the optic axis, supporting the extraordinary wave propagation. The y and z‐axes are the ordinary axes, supporting the ordinary wave propagation.
Figure (4.4) shows a microstrip line of width w on an anisotropic substrate of thickness h. It forms a parallel‐plate capacitor placed in the (x‐z)‐plane. It views εr‖ component of the uniaxial relative permittivity. Whereas, if the parallel plates of the capacitor are placed either in the (x‐y)‐plane or the (y‐z)‐plane, it will view the εr⊥ component of a uniaxial substrate medium. Thus, a uniaxial dielectric medium offers two different values of capacitance, depending on the placement of the parallel plates, connected to a voltage source. Normally, manufacturers provide data for εr‖ and εr⊥ of the uniaxial substrates. The constitutive relation for the uniaxial medium aligned to the physical axes is
(4.2.9)
In the case of the crystal axes ξ, η of an anisotropic substrate, shown in Fig. (4.4), have an angle θ = θ1 = θ2 with respect to the physical axes x, y; the relative permittivity components could be computed by the following expressions [B.24]:
(4.2.10)
Type III: Biaxial materials. For such an anisotropic medium, all three principal relative permittivity components are different, i.e. εrξ ≠ εrη ≠ εrζ. Such a medium is known as the biaxial medium [B.3].
The above nomenclatures are also applicable to the permeability of a magnetic material.
Anti‐symmetric Anisotropic Materials
The plasma medium, i.e. the electron gas model of metal or ionosphere, is treated as an isotropic medium. However, in the presence of a static biasing magnetic field, it becomes a uniaxial dielectric material with off‐diagonal elements for the permittivity matrix. This permittivity matrix is anti‐symmetric. It does not support linearly polarized characteristic waves as the normal modes. It forms a gyroelectric medium, i.e. electrically gyrotropic medium. Similarly, in the presence of a static biased magnetic field in the z‐direction, ferrite medium becomes a gyromagnetic medium, i.e. magnetically gyrotropic medium. Both plasma and ferrite media, under the magnetic biasing, support propagation of circularly polarized characteristic waves as the normal modes [B.2–B.4, B.21, B.22]. The permittivity and permeability tensors in matrix form for these media are summarized below.
Gyroelectric Medium
The relative permittivity matrix for gyroelectric medium, under the applied magnetic field in the z‐direction, is expressed as follows:
(4.22)
In the above expressions, ωp and ωc are the plasma frequency and cyclotron frequency. The cyclotron frequency is also called the gyrofrequency. The jκ is a cross‐coupling factor responsible for the gyration property of the relative permittivity of a medium. Other parameters are N: electron density, e: electron charge, m: electron mass, and B0: DC magnetic field. In the case of the extremely high magnetic field, ωc → ∞ and εr = 1, κ = 0. Also, in the absence of magnetic field, B0 = 0, κ = 0, and the gyrotropic medium is reduced to a uniaxial dielectric medium. In some cases, there is no plasma medium in the z‐direction and εr, zz = 1.
Gyromagnetic Medium
The relative permeability matrix of a gyromagnetic ferrite medium is given below:
(4.2.12)
In the above equations, H0 is the biasing magnetic field in the z‐direction, Ms is the saturation magnetization, and the parameter γ is the gyromagnetic ratio. Again the cross‐coupling factor jκ is responsible for the gyration property of the relative permeability of a medium. The permeability function shows a singularity at the frequency ω = ω0. It is suppressed in the presence of losses. In the case of unbiased and demagnetized ferrite, H0 = 0 and Ms = 0 leading to k = 0 and μr = 1. The ferrite, in this case, acts as a dielectric medium with permeability μ = μ0.
Magneto‐dielectric Composite Materials
The magneto‐dielectric (MD) materials have both electric and magnetic properties. These materials are characterized by the simultaneous presence of permittivity and permeability tensors. These tensors for uniaxial MD materials are expressed as follows:
(4.2.13)
The above‐given expressions for the uncoupled independent relative permittivity and permeability tensors show that there is no coupling between the electric and magnetic fields. Ferrites have such property. However, ferrites are very lossy at microwave frequency. The low‐loss MD materials are synthesized by mixing ferrites/hexaferrite and their composites with a polymer as a host medium [J.3]. The periodic structures are embedded in the host medium to engineer MD materials for antenna applications in VHF and UHF bands [J.4]. The metamaterial composites are also MD materials with simultaneously negative permittivity and permeability in a certain frequency band.
Magnetoelectric Materials
The magnetoelectric materials are general bianisotropic electromagnetic materials with cross‐coupling of electric and magnetic fields. These materials have anisotropy for both the permittivity and permeability with additional cross‐coupling of the electric and magnetic field. In such materials, the electric flux density vector
