Nonlinear Propagation of annular Gaussian beams with different energy in air

The annular Gaussian beams having different initial energies propagation in air are numerically studied by two models, the linear model and the nonlinear model. Numerical results show that (i) at the beginning of the propagation the nonlinear Kerr effect slightly suppresses the focusing caused by the spatial chirp, and (ii) the focusing caused by the spatial chirp generates a high intensity core making the beam shape near the axis similar to the Gaussian shape and this can enhance the Kerr self-focusing, and (iii) the initial energy of the annular beams can influence the nonlinear foci, filamentation length, and the fluence of the propagation. The nonlinear foci decrease with the initial energy, but the foci are not inversely proportional to the square root of the initial peak power, which is not the same as the Gaussian beam. The filamentation length increases with the initial input energy.