Bunching instability of rotating relativistic electron layers and coherent synchrotron radiation

We study the stability of a collision-less, relativistic, finite-strength, cylindrical layer of charged particles in free space by solving the linearized Vlasov-Maxwell equations and compute the power of the emitted electromagnetic waves. The layer is rotating in an external magnetic field parallel to the layer. This system is of interest to understanding the high brightness temperature of pulsars which cannot be explained by an incoherent radiation mechanism. Coherent synchrotron radiation has also been observed recently in bunch compressors used in particle accelerators. We consider equilibrium layers with a "thermal" energy spread and therefore a nonzero radial thickness. The particles interact with their retarded electromagnetic self-fields. The effect of the betatron oscillations is retained. A short azimuthal wavelength instability is found which causes a modulation of the charge and current densities. The growth rate is found to be an increasing function of the azimuthal wave number, a decreasing function of the Lorentz factor, and proportional to the square root of the total number of electrons. We argue that the growth of the unstable perturbation saturates when the trapping frequency of electrons in the wave becomes comparable to the growth rate. Owing to this saturation we can predict the radiation spectrum for a given set of parameters. Our predicted brightness temperatures are proportional to the square of the number of particles and scale by the inverse five-third power of the azimuthal wave number which is in rough accord with the observed spectra of radio pulsars.