Radio ray tracing in complex space

Abstract

When a ray in free space impinges obliquely on the boundary of a lossy medium, whose refractive index is complex, it is refracted at a complex angle, and the coordinates of points on it must take complex values. It follows that, when a ray travels from one real point to another through media some of which are lossy, the ray path can strictly be traced only by using complex values of some of the space coordinates. Approximate treatments using only real space have been used in the past for weakly attenuating media, but this is not accurate for heavier attenuation, and, moreover, there are some problems where the use of complex space is advantageous even for loss free media. The intensity of the received signal is determined partly by attenuation, and partly by convergence or divergence of neighbouring rays. By allowing for this, the convergence of rays near a focus, or a 'skip’ zone can be studied. The technique of using complex space and of allowing for ray convergence is worked out for a plane stratified ionosphere. Examples are given first for an isotropic ionosphere in which electron collisions are not negligible, and then for an anisotropic ionosphere. It is shown how the technique can be used to find the angular diameter of the ‘Ellis window’, and the results obtained agree well with Ellis’s experimentally determined value.