Abstract An entropic formulation to describe the free propagation of Gaussian beams, in a similar way to the thermodynamic theory is developed. We consider two basic applications: (1) an extension to super-Gaussian-like (SGL) beams, and (2) the effect of lenses (convergent and divergent) on the propagation of Gaussian beams. We are interested in such applications because the SGL profiles are obtained through the convolution product using rectangle and Gaussian functions and so, they can be related to the Gaussian beams. The propagation in the Fresnel and Fraunhofer regions are studied, obtaining the laws for the optical entropy of the system. Also, we include some properties and a brief discussion about the condition under which the beam can be considered as an isolated system. For both applications, the evolution of the characteristic width is derived from the entropic postulates.