A method for calculating the "undulator spontaneous emission" produced in waveguides or in free space

We present here a new method to calculate, quite instantaneously, the amplitude and phase transverse distribution of the spontaneous emission (SE) produced in a waveguide. The SE is the radiation produced by an electron beam passing through an undulator. It is involved in the operation of Free Electron Lasers (FEL). Some FELs, working in infrared, use a waveguide in place of the vacuum chamber of the undulator. The numerical process which is described here is quasi analytical. It is based on a standard method for wave propagation using the Fast Fourier Transform, and it is compatible with numerical codes for wave propagation. It is limited to planar undulators. Some numerical codes exist to calculate the SE transverse pro le, but they are not suitable for waveguide configuration. In addition, a simplified version of the calculation gives the SE radiation without waveguide, i.e. when propagating in free space. In practice, the process can be included in a general code to calculate the cavity losses at the start-up of the FEL operation. These losses are always larger than those for the steady state regime, and they may be large enough to forbid laser operation.