Irradiance-variance behavior by numerical simulation for plane-wave and spherical-wave optical propagation through strong turbulence

We have simulated optical propagation through atmospheric turbulence in which the spectrum near the inner scale follows that of Hill and Clifford [J. Opt. Soc. Am. 68, 892 (1978)] and the turbulence strength puts the propagation into the asymptotic strong-fluctuation regime. Analytic predictions for this regime have the form of power laws as a function of β₀², the irradiance variance predicted by weak-fluctuation (Rytov) theory, and l₀, the inner scale. The simulations indeed show power laws for both spherical-wave and plane-wave initial conditions, but the power-law indices are dramatically different from the analytic predictions. Let σ_I²− 1 = a(β₀²/β_c²)^−b(l₀/R_f)^c, where we take the reference value of β₀²to be β_c², = 60.6, because this is the center of our simulation region. For zero inner scale (for which c = 0), the analytic prediction is b = 0.4 and a = 0.17 (0.37) for a plane (spherical) wave. Our simulations for a plane wave give a = 0.234 ± 0.007 and b = 0.50 ± 0.07, and for a spherical wave they give a = 0.58 ± 0.01 and b = 0.65 ± 0.05. For finite inner scale the analytic prediction is b = 1/6, c = 7/18 and a = 0.76 (2.07) for a plane (spherical) wave. We find that to a reasonable approximation the behavior with β₀²and l₀indeed factorizes as predicted, and each part behaves like a power law. However, our simulations for a plane wave give a = 0.57 ± 0.03, b = 0.33 ± 0.03, and c = 0.45 ± 0.06. For spherical waves we find a = 3.3 ± 0.3, b = 0.45 ± 0.05, and c = 0.8 ± 0.1.