Superradiant atomic recoil lasing with orbital-angular-momentum light

We analyze the collective light scattering by cold atoms in free space of a pump-laser beam possessing orbital angular momentum. derive set coupled multiparticle equations for atomic motion which vacuum-mode field is adiabatically eliminated. The resulting describe recoil as due to either transfer linear momentum or For transverse annular distribution initial equilibrium with uniform phases and no scattered unstable. are rotation bunched phase at different harmonics depending on pump azimuthal index $\ensuremath{\ell}$ ring radius.