Exact solution for second harmonic generation in XFELs

The generation of harmonic radiation via a non-linear mechanism, driven by electron bunching at the fundamental frequency, is an important option in the operation of high-gain Free-Electron Lasers (FELs). By utilizing harmonic generation at a large scale facility, the production of intense radiation at shorter wavelengths for the same electron beam energy is feasible. This paper describes a theory of second harmonic generation in planar undulators with particular attention to X-Ray FELs (XFELs). Our study is based on an exact analytical solution of Maxwell’s equations, derived with the help of a Green’s function method. Up-to-date theoretical understanding of second harmonic generation is limited to the estimation of the total radiation power, which is based on a comparison of the right hand side of the wave equation for the first harmonic with the right hand side of the equation for the second harmonic, the latter being incorrectly modified. The exact solution should be obtained by solving the wave equation itself. Our work yields correct parametric dependencies and specific predictions of additional properties such as polarization, angular distribution of the radiation intensity and total power. The most surprising prediction is the presence of a vertically polarized part of the second harmonic radiation, whereas current knowledge predicts a horizontally polarized field.