Electromagnetic propagation in an exponential ionization density

The propagation of a plane electromagnetic (TE) wave into a plane stratified medium in which the ionization density varies as exp (
z/z_{0}
) 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
z = -x
) is given by the simple expression exp
{-(4\pi z_{0}\cos \theta_{i}/\lambda) \tan^{-1}(\nu_{c}/\omega)}
, where
\theta_{i}
is the angle of incidence,
\lambda
is the free-space wavelength and (
\nu_{c}/\omega
) 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.