Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 195

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 ε_zz = ε_‖ and permeability of the medium is μ. The (x‐y)‐plane, with permittivity tensor elements ε_xx = ε_yy = ε_⊥, is a transverse plane. So, the medium is isotropic in the (x‐y)‐plane. The propagation constant along the z‐axis is k_z and in the transverse plane, it is k_t, satisfying the relation . To simplify equation (4.7.19) for the uniaxial medium, the transverse wavevector in the (x‐y)‐plane is aligned such that the wave propagates only along the x‐axis, i.e. k_y = 0 and k_x = k_t. Under such alignment, equation (4.7.19) is reduced to the following dispersion relation:

(4.7.24)

The above equation provides the following characteristic equation:

(4.7.25)

The above expression is also obtained from equation (4.7.19) even for the alignment of the wavevector to the y‐axis, i.e. k_x = 0 and k_y = k_t. So, the characteristic equation (4.7.25) gives the following two independent dispersion relations:

(4.7.26)

The above equations (4.7.26a) and (4.7.26b) demonstrate the presence of two normal modes of wave propagation in the 3D (k_x, k_y, k_z) – 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 . At increasing order of frequencies, equation (4.7.26a) provides isofrequency spherical surfaces. Likewise, equation (4.7.26b) is an equation of ellipsoid in the 3D k‐space at a fixed frequency. It provides the dispersion relation of the extraordinary waves in the form of the concentric isofrequency ellipsoidal surfaces. The wavevector of extraordinary wave k^e is rotation dependent.

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 (k_y‐k_z) plane. It is the dispersion relation of the ordinary waves in the isotropic (y‐z)‐ plane. The wavenumber of the ordinary wave is independent of permittivity ε_‖ along the z‐axis.

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 (k_y‐k_z)‐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 wavevector k^e subtends an angle θ with respect to the z‐optic axis. In this case, wavevectors components are and . Using, these relations with equation (4.7.26d), the following equation of ellipse of permittivity, supporting the extraordinary wave, is obtained:

(4.7.27)

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 n_e 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 . Along the k_y, the extraordinary waves have relative permittivity . Only along the z‐optic axis, i.e. along the k_z‐axis, the identical relative permittivity for both kinds of waves is obtained. In all other directions, the ordinary wave is a fast‐wave as compared to the extraordinary wave. The medium is known as the positive uniaxial medium. Figure (4.15c) shows the case of the negative uniaxial medium. In this case, ε_r‖ < ε_r⊥ and the ordinary wave is a slow‐wave as compared to the extraordinary wave in all directions except along the y‐axis.

The phase velocities v_p1 and v_p2 of the ordinary and extraordinary waves using equations (4.7.26a) and (4.7.27a) are expressed through the following relations [B.3]:

(4.7.28)

The phase velocity v_p1 of the ordinary wave is independent of the angle θ. However, the phase velocity v_p2 of the extraordinary wave is dependent on the angle θ. In Figure (4.15b), both velocities are identical only for the wave propagation along the z‐optic axis.

Finally, it is possible to artificially realize a uniaxial anisotropic material with one of the permittivity components as a negative quantity, say ε_r⊥ = − |ε_r⊥|. In this case, equation (4.7.26d) shows the hyperbolic dispersion relation. This medium is known as the hypermedium. It supports the wave propagation with a larger value of wavenumbers and can convert an incident evanescent wave to the propagating waves. Such engineered materials are needed by the hyperlens [J.1, J.7]. Figure (4.15d) shows the dispersion relation of the hypermedium with phase and group velocities. In this case, y‐components of v_p and v_g are opposite to each other. So, the hypermedium supports the backward wave propagation, and it is a metamaterial medium. The hyperlens is discussed in the subsection (5.5.6) of chapter 5.