Universal scaling law for orbital angular momentum mode purity degradation in Kolmogorov and non-Kolmogorov atmospheric turbulence

The degradation of orbital angular momentum (OAM) modes propagating through atmospheric turbulence fundamentally limits free-space optical communication capacity. We report a universal analytical formula governing this degradation: <i>η</i>_<i>l</i> = [1|(<i>D</i>/<i>r</i>₀/<i>a</i>_<i>l</i>)^5/3]^-1, where <i>a</i>_l=1.64/(∣<i>l</i>∣+1)/2. Based on Kolmogorov turbulence statistics, we derive an analytical integral expression for purity and obtain a closed-form scaling law through Kolmogorov-constrained closure. The exponent 5/3 emerges directly from the Kolmogorov phase structure function, while the mode-dependent coefficient <i>a</i>_<i>l</i> is determined by numerical calibration. We validate our theory against 560,000 independent Monte Carlo simulations spanning topological charges <i>l</i> = 1-10 and turbulence strengths <i>D</i>/<i>r</i>₀ = 0.01-20. The measured exponent <i>b</i> = 1.665 ± 0.042 agrees with the theoretical prediction to within 0.13%, and all fits achieve <i>R</i>² > 0.99. We further extend this framework to non-Kolmogorov turbulence, showing that the exponent generalizes to <i>b</i>(<i>α</i>) = <i>α</i> - 2 for arbitrary spectral index <i>α</i>. A practical lookup table enables rapid link budget estimation without Monte Carlo simulation. These results provide both physical insight into OAM degradation mechanisms and a theoretical foundation for designing high-capacity OAM communication systems.