Numerical Simulation and Analysis of the Scaling Law for Slant-Path Propagation of Laser Beams in Atmospheric Turbulence

Slant-path propagation of laser beams through atmospheric turbulence produces beam spreading and jitter that must be rapidly predicted for system design and performance assessment. Existing scaling laws are mainly derived for horizontal paths and single-parameter variations, which limits their accuracy and applicability to realistic engagement geometries. Here, we construct a comprehensive wave-optics database for 1.064 μm truncated Gaussian beams with a 1 m aperture by traversing initial beam quality factor β0, propagation distance L, elevation angle θ, turbulence strength Cₙ2, and tracking jitter. From 46,800 turbulence-only cases, we extract the 63.2% encircled-power expansion factor and quantify the coupled influence of β0, L, and θ on the turbulence term coefficient A in the scaling expression. A compact 3–10–1 feedforward neural network is trained to map (β0, L, θ) to A, achieving a coefficient of determination R2 = 0.948. Additional simulations without turbulence show that the jitter term coefficient B is nearly invariant over the considered parameter range, with an average value B = 3.69. Combining these results yields a unified scaling law for linear beam spreading on horizontal and slant paths. Comparison with full-wave-optics simulations demonstrates that the proposed law reproduces horizontal-path results and significantly reduces prediction errors at θ = 60° relative to existing models, providing an efficient tool for beam-quality prediction and performance evaluation in atmospheric laser propagation.