Comment on Coherent Electron Cooling
An interesting method proposed in [1] coheren t electron cooling (CEC) of ion beams in a storage ring of the Large Hadron Collider (LHC). The authors est ima ed that theirs method, in contrast to the other  traditional methods of cooling, is an efficient coo ling procedure for the proton beam at high energy. As the subject of beam cooling is important for practi cal applications, we decided to compare the method described in [1] with the one described earlier in [2]. In the method proposed in [1], a uniform cylind rical electron and ion beams pass along the modulat ory section with the same speed. Heavy ions attract the electrons so that at the exit of the modulator sec tion, the electron beam has enhanced charge density aroun d each ion. Sequentially, the electron beam is sepa rated from the ion one and enters the undulator, wh ere each region with its high charge density emits a coherent undulator radiation (CUR). Emitted along t he beam axis, CUR, passing through the front part o f the moving beam, modulates its density and, in turn , this density modulation radiates a CUR superimposed on the previously emitted wave. In this case CUR becomes amplified as well as the density modulation of electron beam. This section is essentiall y free-electron laser (FEL) in a mode of self-amp lified stimulated emission (SASE). As a result, at the out put of the FEL in the direction of its axis CUR has the form of a wave packet (wavelet) with length min , 1 λ K and the electron beam density modulation at the