Line shape of soft photon radiation generated at zero angle in an undulator with a dispersive medium

The problem of undulator radiation from a bunch of charged particles, taking into account a medium polarization, is considered. In a dispersive medium, at a zero angle, in addition to hard photons, soft photons are also generated. If the wavelength of the soft photons is greater than or equal to the longitudinal size of the microbunches formed during the FEL process, the microbunches radiate coherently. Consequently, the radiation of the bunch will be partially coherent. As a result, intense, quasi-monochromatic, and directed X-ray photon beams are produced, which can have wide practical applications.