The increasing number of USUI laser facilities worldwide is driving a pressing need to directly characterize the laser focal parameters. Electron scattering has been widely considered as a promising method for this demand, and its feasibility has been experimentally validated. In the most previous studies, however, the laser parameter (especially the peak intensity) is inferred under the plane-wave approximation, which is fundamentally inadequate for tightly focused USUI laser. Therefore, establishing the scaling relation between the quality of scattered electron and laser parameters over a broad range is an essential requirement for enabling such diagnostics.
In this work we study the scaling laws of electron scattering driven by a relativistic Gaussian laser both with ponderomotive theory and multiparametric test-particle simulations. The electron scattering angle θ_c dependence on three laser parameters, i.e., peak amplitude a₀, spot size w₀, and pulse duration τ₀, is shown to degenerate to a dependence on only two parameters, a₀ and cτ₀=w₀. Furthermore, the electron exhibits three typical kinds of scattering trajectories under different ratios of cτ₀=w₀, each associated with a distinct scaling relation. These theoretic predictions are well confirmed by the multiparametric simulation results, which are used to fit a common scaling relation between the electron scattering angle and the three laser focal parameters. This scaling relation is also validated by compared to the previous experiment results, thus providing a theoretical basis for directly characterizing laser focal parameters under full power by using electron scattering.

超强激光,有质动力,参数表征