Introduction To Modern Planar Transmission Lines. Anand K. Verma

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(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].

      4.7.3 Dispersion Relations in Biaxial Medium

      A biaxial medium could be considered with scalar permeability μ and permittivity tensor [ε]. The off‐diagonal elements of the matrix equation (4.2.4a) are zero. Maxwell equations (4.5.31a) and (4.5.31b) are used in the present case with permittivity tensor [ε] in place of a scalar ε. The wave equation (4.5.32a) is suitably modified to incorporate the tensor [ε]:

(4.7.19)

(4.7.20)

Using equation (4.7.20), equation (4.7.19) is rewritten as,

(4.7.21)

      The nontrivial solution for E_i (i = x, y, z) of the above homogeneous equation is det[ ] = 0, i.e.

(4.7.22)

The above dispersion relation is a quadratic equation of any component of k². So, there are two solutions for any component of k. Two solutions correspond to two normal modes of propagation in the anisotropic medium. At a fixed frequency, equation (4.7.22) is the equation of an ellipsoid surface in the k‐space (wavevector space), i.e.

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