When an electron bunch is accelerated in an ion cavity, the ion cavity has an electric field in not only longitudinal direction but also transverse direction.
Longitudinal part of the electric field will accelerate electrons to a propagation direction of a laser pulse but the transverse component of the electric field attracts the electron bunch to the propagation axis. Therefore, the accelerating electron bunch has oscillation motion in the cavity depends on initial condition such as where the bunch injected into the cavity and how much transverse momentum does it have.
When a charged particle beam hit a metal foil, transition radiation will be generated in an angle inverse proportional to Lorentz factor of the beam.
If a length of the beam is shorter than the wavelength of the radiation, the radiation becomes coherent. Accelerated electron beams from laser wakefield accelerators have a length of few micrometers and it is much smaller than typical transition radiation spectrum of THz regime(~0.1mm), so the radiation should be coherent. Now we are studying about this topic with particle-in-cell (PIC) simulations and we will do some experiment soon.