Electromagnetic propagation in an exponential ionization density

Abstract

The propagation of a plane electromagnetic (TE) wave into a plane stratified medium in which the ionization density varies as exp ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z/z_{0}</tex> ) is investigated. The solution of the wave equation appears as a combination of Bessel functions of imaginary order and complex argument, but the magnitude of the reflection coefficient (taken at <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">z = -x</tex> ) is given by the simple expression exp <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{-(4\pi z_{0}\cos \theta_{i}/\lambda) \tan^{-1}(\nu_{c}/\omega)}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\theta_{i}</tex> is the angle of incidence, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\lambda</tex> is the free-space wavelength and ( <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\nu_{c}/\omega</tex> ) is the ratio of electron collision frequency to the frequency of the field. In general, the field components must be obtained from a complex series, but at depths beyond the critical density asymptotic forms are given which display the rapid decay of the evanescent field in terms of elementary functions.