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/z₀) 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 atz = -x) is given by the simple expression exp{-(4π z₀cos θᵢ/λ) tan⁻¹(νc/ω)}, whereθᵢis the angle of incidence, the free-space wavelength and (νc/ω) 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.