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 slab converts linearly polarized incident waves into the circularly polarized waves. It is shown below:
(4.7.8)
The electric components of the extraordinary and ordinary waves and also the total E‐field at the output of the slab are
(4.7.9)
The wave at the output of the slab is a left‐hand circularly polarized wave. Such a slab converting the incoming linearly polarized wave to the circularly polarized waves is called the quarter‐waveplate. A waveplate with
The above characteristics of a slab are also realized by thin metasurfaces discussed in subsections (22.5) and (22.6) of chapter 22.
4.7.2 Wave Propagation in Uniaxial Gyroelectric Medium
Figure (4.13b) shows uniform TEM‐waves propagation in the z‐direction in an unbounded uniaxial gyroelectric medium created by the magnetized plasma on the application of the DC magnetic field H0 in the z‐direction. The permittivity tensor [εr] of the medium is given equation (4.2.11). Due to the presence of off‐diagonal matrix elements ±jκ in the permittivity matrix of the gyroelectric medium, the Ex component of the linearly polarized incident wave also generates the Ey component with a time quadrature. It is due to the presence of factor “j.” Similarly, the Ey component of an incident wave generates Ex component also with a time quadrature. The presence of two orthogonal E‐field components with a time quadrature in a gyroelectric medium creates the left‐hand circularly polarized (LHCP) and right‐hand circularly polarized (RHCP) waves as the normal modes in the uniaxial gyroelectric medium. Both circularly polarized waves travel with two different phase velocities. Thus, the gyro medium with the cross‐coupling gyrotropic factor ±jκ has the ability of polarization conversion.
Maxwell equation (4.7.2a) is expanded in the usual way to get the transverse field components Ex, Ey and Hx, Hy:
However, the Maxwell equation (4.7.2b) in the present case is expanded differently:
For the uniform plane wave propagating in the positive z‐direction, ∂Hy/∂x = ∂Hx/∂y = 0. In the above equations, it is noted that the εr, zz component of permittivity does not play any role in the TEM mode wave propagation in the z‐direction. However, for the wave propagation in the x‐direction Ex = 0, Ez ≠ 0 and εr, zz permittivity component occurs in the wave propagation. Similar is the case for the wave propagation in the y‐direction. Further, due to the cross‐coupling between Ex and Ey components in the above equations, it is not possible to obtain a single second‐order wave equation for either Ex or Ey. However, the solution could be assumed for the field vectors
