Focusing femtosecond X-ray free-electron laser pulses by refractive lenses

The possibility of using a parabolic refractive lens with initial X-ray free-electron laser (XFEL) pulses, i.e. without a monochromator, is analysed. It is assumed that the measurement time is longer than 0.3 fs, which is the time duration of a coherent pulse (spike). In this case one has to calculate the propagation of a monochromatic wave and then perform an integration of the intensity over the radiation spectrum. Here a general algorithm for calculating the propagation of time-dependent radiation in free space and through various objects is presented. Analytical formulae are derived describing the properties of the monochromatic beam focused by a system of one and two lenses. Computer simulations show that the European XFEL pulses can be focused with maximal efficiency, i.e. as for a monochromatic wave. This occurs even for nanofocusing lenses.