Phase integrals and coupling for radio waves in the ionosphere

Abstract
The phase integral method, for radio waves vertically incident on a horizontally stratified ionosphere, involves a study of the behaviour of the refractive indices in the complex height plane. The two characteristic waves, ordinary and extraordinary, either both upgoing or both downgoing, are said to be critically coupled at certain points in the complex height plane, called coupling points, where the two refractive indices are equal. There are two principal coupling points, and previous authors have considered only the one that lies on the positive imaginary side of the real height axis when collisions are neglected. It is shown here that a full description of the coupling processes also requires the use of the other principal coupling point, which usually lies on the negative imaginary side of the real height axis. Phase integral formulae are derived which describe the changes in polarization, caused by coupling, for the two main ionospheric reflexions, for frequencies less than the gyrofrequency. These formulae are compared with results obtained by a full wave analysis and show good agreement.
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