Kerr effect on propagation characteristics of Hermite-Gaussian beams

The analytical propagation formulae of Hermite-Gaussian (H-G) beams propagating in Kerr media are derived by using the variational approach. The analytical formulae of the self-focusing critical power, the Rayleigh range and the beam quality factor of H-G beams propagating in Kerr media are also derived. It is demonstrated that the ABCD law is valid if a new complex parameter is introduced, which presents a simple method to study the propagation of H-G beams through an optical system in Kerr media. It is shown that, as the beam order increases, the self-focusing critical power of H-G beams increase and H-G beams are less affected by the Kerr nonlinearity. Finally, it is found that the focus point of H-G beams never appears in self-focusing media, and the focal length is not suitable for characterizing the self-focusing effect of H-G beams.