Analytical studies of coherent electron cooling

Under certain assumptions and simplifications, we studied a few physics processes of Coherent Electron Cooling using analytical approach. In the modulation process, the effect due to merging the ion beam with the electron beam is studied under single kick approximation. In the free electron laser (FEL) amplifier, we studied the amplification of the electron density modulation using 1D analytical approach. Both the electron charge density and the phase space density are derived in the frequency domain. The solutions are then transformed into the space domain through Fast Fourier Transformation (FFT)