Generation of coherent radiation in a resonant medium by a relativistic charged particle

Abstract

The resonant radiation due to the coherent excitation of the electric dipole moments of the atoms which are embedded in a host medium with real dielectric constant $\ensuremath{\epsilon}$ has been studied. These dipole moments are induced by the electric field of a relativistic charge particle moving through the resonant medium. In this context, coherent excitation refers to the correlation of phases of the dipole moments, which are induced by a single relativistic charged particle. The phases of dipole moments which are induced by different charged particles are, of course, uncorrelated. The density $\mathfrak{N}$ of the embedded atoms in the host medium is assumed to be sufficiently small so that $\mathfrak{N}|\ensuremath{\alpha}({\ensuremath{\omega}}_{r})|\ensuremath{\ll}1$, where $\ensuremath{\alpha}({\ensuremath{\omega}}_{r})$ is the polarizability of the atoms at the resonance frequency ${\ensuremath{\omega}}_{r}$. Under this assumption, the electric field acting upon each atom is approximately equal to the electric field of the incident relativistic charged particle. This field is analyzed into its frequency spectrum and the effect of each frequency component on the embedded atoms is determined. Expressions for the fields, far away from the resonant medium, and the radiated energy are derived for transitions between bound states. Two cases are considered. In the first case, it is assumed that $\ensuremath{\zeta}={(1\ensuremath{-}\ensuremath{\epsilon}{\ensuremath{\beta}}^{2})}^{\ensuremath{-}\frac{1}{2}}$ is real, where $\ensuremath{\beta}=\frac{v}{c}, v$, being the velocity of the incident charged particle and $c$ the velocity of light. It is shown that if $\ensuremath{\zeta}\ensuremath{\gg}1$, the radiation is emitted in the direction of motion of the incident charged particle. In the second case, it is assumed that $\ensuremath{\zeta}$ is imaginary. Since the \ifmmode \check{C}\else \v{C}\fi{}erenkov condition is satisfied, i.e., $\ensuremath{\beta}\sqrt{\ensuremath{\epsilon}}>1$, there is a primary \ifmmode \check{C}\else \v{C}\fi{}erenkov radiation in the host medium which excites coherently the embedded atoms at their resonance frequency. In spite of the fact that both the primary and stimulated radiations are emitted in the direction of the \ifmmode \check{C}\else \v{C}\fi{}erenkov cone, it is shown that it is possible to detect the presence of the latter because of the finite lifetime of the embedded atoms. Therefore, the possibility arises for the manifestation of their coherent excitation by relativistic charged particles.