The short-term average irradiance profile of a focused laser beam transmitted through a homogeneous-isotropic medium has been determined by using the extended Huygens-Fresnel principle and by modifying the phase structure function to remove tilt. In contrast to previous analysis, no assumption is made regarding the independence of the distribution of phase with tilt removed and the random vector β defining tilt. This analysis applies to the near field of the effective coherent transmitting aperture, where the beam wanders as a whole and does not break up into multiple patches or blobs. Central to the analysis is the short-term average mutual coherence function (MCF) of a spherical wave, which has been determined from the modified phase structure function. Assuming a Kolmogorov spectrum, the modified phase structure function has been determined for three specific aperture functions. These same aperture functions are then used to determine the short-term irradiance profiles. Numerical calculations have been performed, and the results are presented for uniform and Gaussian aperture functions for various values of aperture obscuration and for various strengths of turbulence values. Comparisons are made between the long-term average, short-term average, and Fried’s short-term average irradiance profiles. In particular, on-axis irradiance values and beam spread, as determined by the 1/e points in irradiance, are compared. It is found, in contrast to previous analysis, that the short-term beam spread remains relatively constant as the strength of turbulence becomes large and then increases slowly.
M. T. Tavis and H. T. Yura, "Short-term average irradiance profile of an optical beam in a turbulent medium," Appl. Opt. 15, 2922-2931 (1976)