  We study the stability of a collisionless, 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 non-zero 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 wavenumber, 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
wavenumber which is in rough accord with the observed spectra of radio pulsars.
