Coupling of the Ordinary and Extraordinary Rays in the Ionosphere

Abstract

In this paper an approximate solution to the wave equation is given for propagation in an ionosphere in which the gradient of the density N is in the vertical, z, direction only, and in which account is taken of the earth's magnetic field. It corresponds exactly to the ray theory and expresses a quantity Z, which is the z derivative of the phase function S, by a quartic equation. Z can be represented as a function of ζ (which is proportional to N) on a four-sheeted Riemann surface, and the branch points are studied for the case of vertical incidence for which Z becomes the refractive index. By considering the branch points in the complex ζ plane, the amount of the coupling between the ordinary and the isolated extraordinary branches of the (Z, ζ) curves can be expressed as a function of the obliquity of the magnetic field. The triple splitting of rays reflected from the ionosphere, observable where the field is nearly vertical, can thus be explained, and the theory is substantiated by the observation that the polarizations of the echoes on the (P', f) records are ordinary, ordinary and extraordinary in order of increasing critical frequencies, as given by the branches of the (Z, ζ) curves.