Introduction To Modern Planar Transmission Lines. Anand K. Verma
rel="nofollow" href="#ulink_7e479a52-2846-5796-a7a6-0589958bb843">equation (4.6.21a).
4.7 EM‐waves Propagation in Unbounded Anisotropic Medium
Two cases of wave propagations in the uniaxial anisotropic media – without off‐diagonal elements and with off‐diagonal elements, are considered in this section. The dispersion relation is also discussed leading to the concept of hypermedia useful for the realization of hyperlens [J.1, J.5–J.7].
4.7.1 Wave Propagation in Uniaxial Medium
The unbounded lossless homogeneous uniaxial medium is considered. The y‐axis is the optical axis, i.e. the extraordinary axis. In the direction of the optic axis, the permittivity is different as compared to the other two directions. The medium is described by a diagonalized matrix with all off‐diagonal elements zero. The permeability of the medium is μ0 and its permittivity tensor is expressed as follows:
(4.7.1)
Figure (4.13a) shows the TEM plane wave propagation in the x‐direction. The TEM waves have Ex = 0, Hx = 0; Ey ≠ 0, Hy ≠ 0; Ez ≠ 0, Hz ≠ 0. Also, the uniform field components in y and z‐directions do not vary, i.e. ∂/∂y(field) = ∂/∂z(field) = 0. Under these conditions, the following Maxwell equations provide the transverse field components of electric and magnetic fields:
On expansion, the above equations provide the following sets of transverse field components:
Figure 4.13 Wave propagation uniaxial media.
On eliminating Hz and Hy from the above equations, wave equations for the electric field transverse components are obtained. Likewise, the wave equations for magnetic field transverse components are obtained on eliminating Ez and Ey:
Equation (4.7.5a) is a 1D wave equation of the y‐polarized electric field Ey ≠ 0 and Ez = 0. The magnetic field component Hz is along the z‐axis. The Ey field component causes polarization in the dielectric medium creating relative permittivity εr‖ along the y‐axis. Further, the electric field Ey is transverse to the (x‐z)‐plane containing the direction of propagation x. Such waves are called the transverse electric (TE) waves. The z‐polarized electric field with Ez ≠ 0 and Ey = 0 follows the wave equation (4.7.5b). The Ez component generates another polarization in the dielectric medium creating relative permittivity εr⊥ along the z‐axis. In this case, Hy‐component is transverse to the (x‐z)‐plane. These waves are called the transverse magnetic (TM) waves. In the present case, also in the case of the oblique incident of the plane waves, the waves are still TEM only. However, the TE and TM terminology is normally used for the non‐TEM mode waves supported by the waveguides. It is discussed in chapter 7. A reader should be careful in the dual use of terminology TE/TM to show the mode of propagation, and also the polarization of propagation. For the guided waves, TE/TM is modes of propagation; and in the open medium, it indicates the polarization of the propagating waves. The second use in the context of obliquely incident EM‐waves at the interface of two medias is further discussed in section (5.2) of chapter 5.
On solving the above wave equations, the time‐harmonic wave propagating in the positive x‐direction is obtained as,
In the above equations, k0 and c are wavenumber and velocity of EM‐waves in free space. Also, ke and ko are wavenumbers of the extraordinary waves and ordinary waves, respectively, traveling in the x‐direction with phase velocities vpe and vpo. The extraordinary waves are y‐polarized, i.e. TE‐polarized wave viewing the permittivity component εr‖. The ordinary waves are z‐polarized, i.e. TM‐polarized wave viewing the permittivity component εr⊥. Thus, an obliquely incident linearly polarized EM‐waves, with Ey and Ez components, entering the slab of the anisotropic medium is split into two distinct normal mode waves and travel with two different phase velocities. They come out from the slab with a phase difference. This phenomenon is known as double refraction or birefringence. The dispersion relation for both normal waves is discussed in subsection (4.7.5).
The wave impedances ηe and ηo of the extraordinary waves and ordinary waves propagating in the x‐direction are obtained by substituting the field solutions of equation (4.7.6) in equations (4.7.3) and (4.7.4):
(4.7.7)
The plasma medium could be taken as an example. It is a uniaxial anisotropic medium with εr⊥ = 1 and εr‖ = εr. In this case, y‐polarized extraordinary waves travel with a slower phase velocity vpe as compared to a phase velocity vpo of the z‐polarized ordinary waves. So, the extraordinary waves are also known as the slow‐waves with εr‖ > εr⊥. The ordinary waves are called fast‐waves. The optic axis of the uniaxial medium is called the slow‐wave axis and ordinary axis as the fast‐wave axis.
Waveplates and Phase Shifters
The incident linearly polarized wave on a slab at 45° has two in‐phase E‐field components. After traveling a distance d, the field components develop a phase difference Δφ. So, for the case Eoy = Eoz = E0 and a phase difference Δφ = 90° at the output of the slab of thickness x = d, the uniaxial anisotropic dielectric