Atomic diffraction by a thin phase grating

We present a semiclassical perturbation method for the description of atomic diffraction by a weakly modulated potential. It proceeds in a way similar to the treatment of light diffraction by a thin phase grating, and consists in calculating the atomic wavefunction by mean~ of action integrals along the classical trajectories of the atoms in the absence of the modulated part of the potential. The capabilitie~ and the validity condition of'the method are illustrated on the well-known case of atomic diffraction by a Gaussian standing wave. We prove that in this situation the perturbation method is equivalent to the Raman-Nath approximation, and we point out that the usually-considered Raman-Nath validity condition can lead to inaccuracies in the evaluation of the phases of the diffraction amplitudes. The method i~ also applied to the case of an evanescent wave reflection grating, and an analytical expression for the diffraction pattern at any incidence angle is obtained for the first time. Finally, the application of the method to other situations is briefly