Читать книгу Introduction To Modern Planar Transmission Lines - Anand K. Verma - Страница 193

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. 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).