Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 190

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 E_x = 0, H_x = 0; E_y ≠ 0, H_y ≠ 0; E_z ≠ 0, H_z ≠ 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:

(4.7.2)

On expansion, the above equations provide the following sets of transverse field components:

(4.7.3)

(4.7.4)

Figure 4.13 Wave propagation uniaxial media.

On eliminating H_z and H_y 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 E_z and E_y:

(4.7.5)

Equation (4.7.5a) is a 1D wave equation of the y‐polarized electric field E_y ≠ 0 and E_z = 0. The magnetic field component H_z is along the z‐axis. The E_y field component causes polarization in the dielectric medium creating relative permittivity ε_r‖ along the y‐axis. Further, the electric field E_y 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 E_z ≠ 0 and E_y = 0 follows the wave equation (4.7.5b). The E_z component generates another polarization in the dielectric medium creating relative permittivity ε_r⊥ along the z‐axis. In this case, H_y‐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,

(4.7.6)

In the above equations, k₀ and c are wavenumber and velocity of EM‐waves in free space. Also, k^e and k^o are wavenumbers of the extraordinary waves and ordinary waves, respectively, traveling in the x‐direction with phase velocities v_pe and v_po. The extraordinary waves are y‐polarized, i.e. TE‐polarized wave viewing the permittivity component ε_r‖. The ordinary waves are z‐polarized, i.e. TM‐polarized wave viewing the permittivity component ε_r⊥. Thus, an obliquely incident linearly polarized EM‐waves, with E_y and E_z 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 η^e and η^o 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 ε_r⊥ = 1 and ε_r‖ = ε_r. In this case, y‐polarized extraordinary waves travel with a slower phase velocity v_pe as compared to a phase velocity v_po of the z‐polarized ordinary waves. So, the extraordinary waves are also known as the slow‐waves with ε_r‖ > ε_r⊥. 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.