In this paper, the authors have presented a paraxial theory for propagation of (1) Gaussian (2) dark hollow Gaussian high power laser beams in the atmosphere, considering the nonlinearity arising from the temperature variation along the wave-front. Specifically, the focusing parameter for both beams has been evaluated as a function of distance and initial beam power and width (corresponding to radiation of wavelengths 1.045 µ, 1.625 µ, and 2.141 µ in the water absorption window) for the maritime, desert, rural, and urban environments as modeled at NRL; the results have been presented in the dimensionless form. It is seen that in all four environments a dark hollow beam defocuses less than the corresponding Gaussian beam of same radius and power. It is suggested that this conclusion based on the paraxial theory be verified by numerical simulation.
