I do not think that the collimation or tight beamwidth has anything to do with the Doppler effect. The Doppler effect would affect the observed frequency of the radiation. What happens is that if the electrons are moving at near relativistic speeds or greater, then the emitted radiation is sent out in within a small angle, like a spot light, along the electron's velocity vector. Jackson's Classical Electrodynamics text (which I do not have in front of me at home) has a section that deals with this. In fact, if you were to watch a synchrotron from the side, you would see a flash of light/radiation as the electron's beam swept your eye. The faster the electron travels, the tighter the beam becomes. At slower speeds, the beam splits into two beams that emit at an angle away from the electron's velocity vector. Without having Jackson with me to refresh your memory I cannot go into the exact details though.