Quantitative description of thermal self-interaction of laser beams

The relative change in the mean square of the beam radius at asymptotically large distances is used as the integrated characteristic of thermal self-interaction of a laser beam. A description based on the parabolic wave equation is used to derive an equation for this quantity in the form of a sum of two terms, one linear and the other quadratic with respect to the phase change induced by the thermal self-interaction. A Gaussian beam is considered as an example.