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Two-step laser acceleration of protons with two foils and two laser pulses is modelled and optimized. It is shown that a nearly mono-energetic distribution of proton bunches can be realized by a suitable parameter choice. Two-step acceleration schemes make it possible to obtain both higher efficiency and energy as compared to the acceleration with only one laser pulse of an energy equal to the sum of the energy of the two pulses. With the aid of our analytical model, the optimal distance between the two targets, the delay between the two laser pulses, and the parameters of the laser pulses are determined. Estimates and results of the modelling are proven with 2D PIC simulations of the acceleration of proton bunches moving through the second target.


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