An integral for obtaining exactly the electromagnetic field transmitted through a plasma slab having specified electron density-collision frequency profiles

The reflection coefficient of a nonhomogeneous plasma slab satisfies the Riccati differential equation. When the differential equation for the transmission coefficient is combined with the Riccati equation, the resulting expression is analytically integrable. It is demonstrated that transmission of an electromagnetic field through the slab may be expressed entirely in terms of an integral involving the complex wavenumber and reflection coefficient of the medium. It is shown that the exact theory presented here yields correct results when used to obtain the transfer function for a homogeneous imperfectly conducting sheet.