Introduction To Modern Planar Transmission Lines. Anand K. Verma
4.7.5 Dispersion Relations in Uniaxial Medium
This section considers the dispersion relation of a uniaxial anisotropic permittivity medium as a special case of the dispersion relation (4.7.19) of the biaxial medium. The permittivity along the optic axis, i.e. the z‐axis is εzz = ε‖ and permeability of the medium is μ. The (x‐y)‐plane, with permittivity tensor elements εxx = εyy = ε⊥, is a transverse plane. So, the medium is isotropic in the (x‐y)‐plane. The propagation constant along the z‐axis is kz and in the transverse plane, it is kt, satisfying the relation
(4.7.24)
The above equation provides the following characteristic equation:
The above expression is also obtained from equation (4.7.19) even for the alignment of the wavevector
The above equations (4.7.26a) and (4.7.26b) demonstrate the presence of two normal modes of wave propagation in the 3D (kx, ky, kz) – space. Next, the 3D dispersion relation is reduced to the 2D dispersion relation given by equations (4.7.26c) and (4.7.26d). Equation (4.7.26a) is an equation of sphere in the 3D k‐space at a fixed frequency, i.e. at ω = constant. It shows the dispersion relation of the ordinary waves with the wavevector
The 2D dispersion diagram of both the ordinary and extraordinary waves is considered in the (y‐z)‐plane with the help of equations (4.7.26c) and (4.7.26d). Equation (4.7.26c) is the equation of an isofrequency dispersion circle, shown in Fig. (4.15a) in the (ky‐kz) plane. It is the dispersion relation of the ordinary waves in the isotropic (y‐z)‐ plane. The wavenumber
Figure 4.15 Dispersion diagrams in the uniaxial anisotropic medium.
Likewise, equation (4.7.26d) is the dispersion relation of the extraordinary waves, giving the isofrequency ellipse in the (ky‐kz)‐plane, as shown in Fig. (4.15b). In this case, the wavenumber is direction‐dependent because the relative permittivity for this case is direction‐dependent,
The dispersion relation (4.7.27b) is in the term of refractive indices. The refractive index n(θ) is the direction‐dependent, and n0 and ne are refractive indices for the ordinary and extraordinary waves. The wave analysis using the refractive index, in place of permittivity and permeability, is commonly used for the optical wave propagation in the uniaxial medium [B.18–B.20].
Figure (4.15a) shows that the ordinary waves have both the phase and group velocities, on the isofrequency contour, in the same direction. Figure (4.15b) shows the elliptical dispersion diagram for the extraordinary waves. In this case, there is a deviation of the direction of the group velocity from the direction of the phase velocity. The inner circle of Fig. (4.15b) shows the dispersion diagram of the ordinary waves with relative permittivity