Interaction of relativistic electrons with intense electromagnetic fields: ponderomotive\,effect, acceleration,\,refraction,\,reflection, dependence\,on\,initial\,conditions

The rigorous theory and characterization of charged-particle dynamics in high-intensity electromagnetic fields are fundamental for the development of advanced plasma-based applications. Accurate analytical models must bridge the gap between smoothed trajectories and exact particle motion to predetermine injection and energy gain. The main objective of this review is to establish a rigorous framework for the averaged relativistic motion of electrons, focusing on the strict derivation of ponderomotive forces and the impact of fast-oscillating periodic additions on dynamical variables. By making use of the Krylov--Bogoliubov--Mitropolsky averaging method to obtain the equations of motion, the study analyzes relativistic effects in laser beams and waveguides. These theoretical predictions are substantiated through numerical validation, including test-particle simulations and three-dimensional particle-in-cell simulations (PIC) of relativistic self-trapping regimes such as “laser bullet” and “bubble” structures. The review details the independence of the results on the formulation framework, the strict dependence on wave polarization, and the non-strict potential character of the relativistic ponderomotive force. The analysis demonstrates that periodic fast-oscillating additions are essential for a complete description, accurately setting initial conditions in averaged equations and enabling precise predictions of electron reflection and refraction. Simulations confirm that these fast-oscillating corrections determine electron injection and beam charge in realistic laser–plasma acceleration scenarios. The present review clearly shows that the dual framework of test-particle and PIC models is vital for probing the limits of averaged motion theory. The findings are of direct practical relevance for the optimization of radiation sources and guide the development of future theories incorporating non-adiabatic and field topology dependent effects.