Parametric x-ray FEL operating with external Bragg reflectors

In the crystal X-ray FELs using channeling and parametric quasi-Cherenkov mechanisms of spontaneous radiation were considered as versions of FEL allowing, in principle, to obtain coherent X-ray source. In this case a crystal is both radiator and resonator for X-rays emitted by a particle beam passing through crystal. However, it is well-known that a beam current density required for lasing is extremely high in X-ray spectral range for any radiation mechanisms and it is very important to find a way to lower its magnitude. The application of three-dimensional distributed feedback formed by dynamical diffraction of emitted photons permitted to reduce starting beam current density 102-104 times up to 109. One of ways to lower the starting current is the formation of multi-wave distributed feedback the another one is the application of external reflectors. The thing is that lasing regime was shown to be produced at frequencies in the vicinity of degeneration point for roots of dispersion equation describing radiation modes excited in an active medium (crystal plus particle beam). Unfortunately, in case of parametric quasi-Cherenkov FEL this region coincides with the region of strong self-absorption of radiation inside a crystal. That fact, obviously, increases the starting beam current. In this report we have shown that the application of external Bragg reflectors gives the possibility to lower radiation self-absorption inside a crystal by modifying radiation modes excited in the active medium under consideration. The corresponding dispersion equation and the expression for excited modes are derived. The generation equation determining starting conditions for lasing is obtained. Using these expressions we have shown that the application of external Bragg reflectors permits to reduce starting beam current density more than 10 times