Electromagnetic Waves in Stratified and Anisotropic Media – Some Contributions to the Theory

Abstract

In this thesis, three main aspects of the theory of electromagnetic waves in stratified and anisotropic media are investigated. Part I deals with some approximations to the electromagnetic properties of ionized media. First the complex refractive index n is considered for a non-magnetized medium when Z>> 1, Z being the ratio of the electron collision frequency to the angular frequency of the waves. An approximate expression, used in studies of VLF propagation, is examined. It is shown that the conditions for its validity differ from the condition often stated. The possible significance of this for VLF propagation is briefly considered by using a stratified model of the lower ionosphere. Next the complex refractive index n and complex wave polarization R of the magneto-ionic theory are studied. It is found that for sufficiently large Z, the absorption is unaffected by the magnetic field while Re(n) depends only on the longitudinal component of the field. Results are obtained for R when propagation is not purely transverse (where R is always zero or infinity) for large and small Z. For sufficiently large Z the polarization is approximately circular, as it is in the quasi-longitudinal approximation. Part II is concerned with some second order linear ordinary differential equations which arise in various propagation problems involving stratified media. Attention is given to obtaining exact solutions relating to various profiles of the propagation media. First, solutions are obtained for both of the independent 'magnetic' and 'electric' type fields in both planar and spherically stratified media. Attention is then given to propagation transverse to the magnetostatic field in a horizontally stratified, horizontally magnetized ionized medium. Vertically polarized ('electric' type) fields are described by a differential equation involving two elements of the permittivity tensor. A method of transforming this equation is given, which reduces the problem of solving it to one of performing a single integration. By this method a wide variety of plasma profiles and corresponding exact solutions can be generated. Some particular examples of the method are given. In these, it is shown how the forms of some arbitrary parameters can be chosen so that the resulting plasma profiles do not depend on the propagation constant of the waves. Finally in Part II, propagation in cylindrically stratified media is studied for the case in which the fields are independent of distance along the axis of symmetry. Then two independent fields can exist. Some exact wave functions are obtained for both. Part III is concerned with the reflection of waves in isotropic stratified media having profiles which are everywhere continuous. Particular attention is given to profiles which tend to the same constant on either side of a region of rapid variation. Previous studies of such layers have concerned profiles symmetric about an extreme value. When the refractive index deviation is small an approximate integral formula for the reflection coefficient, applicable to both polarizations, can be used. Here, this formula is applied to certain profiles for which the integral can be evaluated analytically, thus giving closed form approximations for the reflection coefficients. The results can be adapted to reflection from isotropic underdense plasma layers by using some approximate single-scattering reflection coefficient formulas developed by Heading. Symmetric profiles and an asymmetric profile are considered. The first order effect of a slight asymmetry on the peak of the profile concerned is to shift the peak without altering the peak value. The slight asymmetry is found to have a first order effect on the magnitude of the reflection coefficient. This cannot be accounted for just by the change in position of the peak. The interpretation is that the effect of the asymmetry in regions away from the peak is important in the reflection process. Also in Part III, the hypergeometric equation is transformed into the wave equation by a transformation more general than that used in the well-known work of Epstein. Thus a more general class of profiles is dealt with. The resulting reflection and transmission coefficients, applicable for vertically incident waves, are obtained by using the connection formulas of the hypergeometric function. In particular, a profile in which the refractive index tends to the same constant on either side of a region of variation is studied. This layer is in general asymmetric, the symmetric Epstein profile being a special case. The case of a slight asymmetry is considered and a simple approximate formula for the reflection coefficient is obtained.