Introduction To Modern Planar Transmission Lines. Anand K. Verma

Introduction To Modern Planar Transmission Lines

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]:

(4.7.19) equation

(4.7.20) equation

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

(4.7.21) equation

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

(4.7.22) equation

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. the normal space. For an isotropic medium, it is reduced to a sphere. Further, on knowing the wavenumber, the field components E_i (i = x, y, z) can be determined from equation (4.7.21).

4.7.4 Concept of Isofrequency Contours and Isofrequency Surfaces

The discussion of dispersion in the uniaxial medium requires an understanding of the concept of isofrequency contours and isofrequency surfaces in the 2D and 3D k‐space. Figure (4.14a–d) explain the concept of isofrequency contours.

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 (k_x‐k_y)‐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 v_p. The direction of the normal at point P shows the direction of the Poynting vector, i.e. the direction of the group velocity v_g. For an isotropic medium, both the v_p and v_g 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.

Schematic illustration of dispersion diagrams of the wave propagating in the z-direction in the isotropic medium.

Figure 4.14 Dispersion diagrams of the wave propagating in the z‐direction in the isotropic medium.

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