Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 192

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 [ε_r] 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 E_x component of the linearly polarized incident wave also generates the E_y component with a time quadrature. It is due to the presence of factor “j.” Similarly, the E_y component of an incident wave generates E_x 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 E_x, E_y and H_x, H_y:

(4.7.10)

However, the Maxwell equation (4.7.2b) in the present case is expanded differently:

(4.7.11)

For the uniform plane wave propagating in the positive z‐direction, ∂H_y/∂x = ∂H_x/∂y = 0. In the above equations, it is noted that the ε_{r, 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 E_x = 0, E_z ≠ 0 and ε_{r, 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 E_x and E_y components in the above equations, it is not possible to obtain a single second‐order wave equation for either E_x or E_y. However, the solution could be assumed for the field vectors and as follows:

(4.7.12)

On substituting the above equations in equations (4.7.10) and (4.7.11), the following sets of equations are obtained:

(4.7.13)

On solving the above equations for E_x and E_y, the following characteristics equation is obtained:

(4.7.14)

where wavenumber in free space is . The det[ ] = 0 of the above homogeneous equation provides the nontrivial solutions giving the following two eigenvalues of the propagation constant β_z:

(4.7.15)

It is shown below that the eigenvalue and are the propagation constant of two circularly polarized normal mode waves propagating in the z‐direction. The wave with propagation constant travels at slower velocity compared to the wave traveling in an isotropic medium with relative permittivity ε_r. The wave with propagation constant is a faster traveling wave.

The electric fields, i.e. the eigenvectors and for both the normal mode waves are obtained by substituting the eigenvalue and in equation (4.7.14), and using equation (4.7.12a):

(4.7.16)

Using equation (4.7.16a) shows the RHCP waves coming toward an observer. Likewise, equation (4.7.16b) is for the LHCP waves coming toward an observer.

Suppose the x‐polarized wave with is incident on the gyroelectric slab of thickness d. At the plane of entry, the linearly polarized electric field can be decomposed into the RHCP and LHCP waves traveling in the positive z‐direction. The electric field at any distance inside the slab is a sum of two circularly polarized waves:

(4.7.17)

However, the wave is still linearly polarized with a rotation of φ with respect to the x‐axis. The angle of rotation φ at the output of the slab is

(4.7.18)

The above equation shows that the E‐field polarization vector rotates while the wave travels in the medium. For the wave reflected at the end of the slab, the total rotation at the input is 2φ. This is known as Faraday rotation. It is the characteristic of a gyrotropic medium – gyroelectric, as well as gyromagnetic [B.2–B.4]. The wave propagation in the gyromagnetic medium is obtained similarly [B.3]. Similar to the gyroelectric medium, the gyromagnetic medium also supports the circularly polarized normal modes. The word gyro indicates rotation and the gyro media supports circularly polarized normal mode wave propagation. They do not support the linearly polarized EM‐waves. The analysis of the wave propagation in other complex media‐ bi‐isotropic and bianisotropic is cumbersome. However, it can be followed by consulting more advanced textbooks [B.13, B.17, B.21–B.23].