Discrete spectrum radiation from a charged particle moving in a medium with Maxwell's fish-eye refraction-index profile

Radiation from a charged particle moving in a medium with a Maxwell's fish-eye refraction index profile is considered. It is shown that the radiation spectrum has a discrete character. The main emitted wavelength is proportional to the refractive profile's radius and has a dipole character in a regular medium. A Cherenkov-like threshold velocity is established. A cardinal rearrangement of angular distribution in a lossless medium is predicted. This behavior is caused by the total internal reflection in a lossless medium as opposed to photons' attenuated total reflection in the regular medium. A lossless medium ensures that both directed and monochromatic emission can serve as a light source in the corresponding regions.