  The stationary eigenstates and eigenvalues for the ponderomotive potential of
an optical crystal confined in a one-dimensional infinite square well are
numerically obtained. The initial states of the incoming particles taken as
Gaussian, are expanded in the basis of the stationary eigenstates of the
ponderomotive potential, to obtain the subsequent time evolution of the wave
function of the particle during the interaction with the optical crystal. From
the results of the time evolution of the probability density, it is observed
that the particles get localized at equidistant positions in the transverse
direction, which results in the diffraction pattern. The temporal evolution of
the diffraction pattern is analyzed. As an application, the diffraction of
proton beams is studied, where the experimental parameters are optimized to
observe the diffraction pattern for a microwave plasma-based proton beam
system. The observations are important for design of proton based matter-wave
interferometers.
