Beam Propagation Factor of Elegant Hermite-Gaussian Beams Propagating through Gain and Absorbing Media

The analytical expression for the field distribution of elegant Hermite-Gaussian (EHG) beams propagating in linear gain and absorbing media is derived. Simple analytical formulas for the far-field divergence and the waist width of EHG beams after passing through gain and absorbing media are also obtained. Furthermore, it can be shown that the beam propagation factor (M² factor) of EHG beams after passing through gain and absorbing media remains invariant under the paraxial approximation.