Semianalytical theory of focusing synchrotron radiation by an arbitrary system of parabolic refracting lenses and the problem of nano-focusing

It has been shown that an arbitrary system of parabolic refracting lenses does not change the shape of the Gaussian wave function of a synchrotron radiation beam. Only three parameters of the wave function change, for which the recurrent formulas are derived. These formulas allow one to perform quickly and accurately the calculation of optical properties of an arbitrary system of lenses. The parameters of the radiation beam calculated by the developed method have been compared to the results of the theory of continuously refracting lens. It has been shown that both approaches give surprisingly close values. The problem of focusing the beam to a nanometer size is discussed. A two-lens system has been proposed, which can provide a five-fold increase in the number of photons inside the focus, although we failed to decrease the beam size.