Fundamentals of Plasma Physics 2025, 27 - Dispersion Relations and Parameter Space

When waves are excited by an antenna in a plane with transmitter frequency ω its geometry imposes the component of the wave-vector in the antenna plane. Assume the case of the x = 0 plane, consequently generating an e^ik_zz dependency. For finite and not perfectly periodic antennae, this creates a spectrum of k_z modes. Solving the mode equations with n_x² = n²sin²θ for both the large and the small roots for arbitrary θ and ω yields that n_x² is only infinite for S = 0. Those regions, where it's complex are inaccessible.

At high frequency waves, the dielectric tensor elements are approximated to

The ion effects have been dropped. For quasi-transverse (θ ≅ π/2) propagation the first term in Γ dominates, so n² can be approximated through a binomial expansion. The + case gives the ordinary case, the - case gives the quasi-transverse-extraordinary (QTX) mode. For quasi-longitudinal (θ ≅ π), the cos²θ terms dominate, and only the leading term of Γ is relevant. In a similar way, two modes emerge for the two different sign cases.