Free space optical (FSO) wireless communication has emerged as a viable alternative to the existing fiber optics and radio frequency (RF) communications due to its ability to operate in unlicensed spectrum, offering huge data handling capacities and being cost effective with easy deployment. The underlying communication mechanism in the FSO link is based upon the laser beam propagation model. The propagating free space optical beam undergoes signal degradation due to refraction and diffraction caused by atmospheric turbulence measured by refractive index structure parameter (Cn2). In this work, a novel, robust analytical model for propagating Gaussian-beam through atmospheric turbulence is presented, by solving the space-fractional paraxial wave equation. We report analytical expressions for the intensity and long-term beam spreading of a Gaussian beam in terms of space-fractional parameter D, for the range 2 < D ≤ 3. This range of parameter D, defines the effective number of euclidean space corresponding to atmospheric turbulence levels faced by the propagating Gaussian beam in the FSO link. The results of the proposed fractional model and the existing models agreed well, therefore the D-dimensional parameter can be used to effectively express the value of refractive index structure parameter (Cn2). The classical values of Cn2 ranges form 10−13 to 10−16 for strong to weak fluctuations respectively. However, in fractional-dimension scale, these fluctuations can be express as D = 2.668 (strong fluctuations) to D = 2.999 (weak fluctuations). The ideal case of free space beam propagation can be expressed as D = 3, with no turbulence. Moreover, we have studied the fractional-dimension model performance for varying wavelengths. Further, based upon a practical example, we proposed a self-sustainable FSO link architecture to efficiently minimize the effect of weak and strong fluctuations on the propagating optical beam, based upon the D-dimension fractional parameter, hence ensuring reliability of FSO link.