Introduction To Modern Planar Transmission Lines - Anand K. Verma

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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)equation

      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)equation

      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]:

equation

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

      4.7.4 Concept of Isofrequency Contours and Isofrequency Surfaces

      The wave propagation is considered in the isotropic (x‐y)‐plane. The dispersion relations for the 2D waves propagation in the isotropic medium and also in air medium are expressed as

      (4.7.23)equation

      At a fixed frequency ω, the above equations are equations of circles in the (kx‐ky)‐plane. Figure (4.14a) shows that the radius of the circle increases with an increase in frequency. It forms a light cone. Figure (4.14b) further shows the increase in the 2D wavevector at the increasing order of frequencies ω1 < ω2 < ω3 < ω4. The concentric contours of the wavevector are known as the isofrequency contours, displaying the dispersion relation. The light cone of the 2D dispersion diagram is generated by revolving Fig. (2.4) of the 1D dispersion diagram, shown in chapter 2, around the ω‐axis. Likewise, 3D isofrequency surfaces are obtained. It is discussed in the next section.

      The propagating wave in the z‐direction is described by equation (4.7.12). The propagation constant images of the waves must be real. So, the propagating waves are obtained only for images, i.e. within the light cone. Outside the light cone, i.e. for images, the waves are nonpropagating evanescent waves. Thus, the light cone divides the k‐space into the propagating and evanescent wave regions shown in Fig. (4.14a,c and d).

      Further, any point P on the isofrequency contour surface, shown in Fig. (4.14b), connected to the origin O shows the direction of the wavevector. It is also the direction of the phase velocity vp. The direction of the normal at point P shows the direction of the Poynting vector, i.e. the direction of the group velocity vg. For an isotropic medium, both the vp and vg are in the same direction. It is noted that the wavefront is always normal to the wavevector images. However, for the anisotropic medium, the phase and group velocities may not in the same direction. It is discussed below.

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