C̆erenkov radiation at the resonance frequency of an amplifying medium

Abstract

The propagation of C\ifmmode \breve{}\else \u{}\fi{}erenkov radiation through a medium in which the population of two energy levels is inverted has been studied. The population inversion is represented by an imaginary part of the index of refraction of the medium at the resonance frequency. The resonant atoms are embedded in a host medium which has the shape of a slab of finite width $L$ and dielectric constant ${\ensuremath{\epsilon}}_{h}$, which is real and varies slowly for frequencies close to the resonance frequency of the embedded atoms. It is assumed that the host medium and the embedded atoms inside the slab behave as an isotropic noncrystalline medium. The slab is covered by a mirror on each of its surfaces and the medium outside the slab and the mirrors has dielectric constant ${\ensuremath{\epsilon}}_{1}$ (${\ensuremath{\epsilon}}_{1}=1$ for vacuum). While each individual charged particle can stimulate coherent emission by different resonant atoms, the phase of radiation which is due to different particles is uncorrelated. It is shown, however, that radiation of high intensity in the direction of the C\ifmmode \breve{}\else \u{}\fi{}erenkov cone can be produced by highcurrent particle beams. Exact expressions, in integral form, are obtained for the electromagnetic field everywhere in space. Using the method of stationary phase approximate expressions are derived for the C\ifmmode \breve{}\else \u{}\fi{}erenkov fields as well as the radiated energy in the foward and backward direction, far away from the slab.