The nonparaxial propagation of the TEM0l doughnut beam with the orbital angular momentum in the far field

The analytical expression for nonparaxial electric field amplitude of the TEM_0l^? doughnut beam with the orbital angular momentum quantum number l is derived in the far field by means of the angular spectrum representation and the stationary phase method. It is shown that the divergent angle of the far field of TEM_0l^? doughnut beam will be smaller with the decreasing of the orbital angular momentum quantum number l or the increasing of the beam waist width w of the initial beam. And the maximal radial intensity of the beam is decreased with its propagation at different rates for different l and w.