  Using a quantum fluid model, a set of three paraxial coupled equations to
describe the FEL instability is derived. These equations are solved numerically
considering Laguerre-Gaussian modes as the initial conditions to study the
transfer of orbital angular momentum (OAM) between them, which satisfies the
total angular momentum conservation as matching condition. The amplification
and compression of the output radiation is observed and using the separation of
variables method, the set of coupled equations, which describes a $\pi$-pulse
solution, is obtained in such a way that a transverse overlapping factor, R,
governs the energy exchange efficiency of the process.
