Approximated analytical solution of the Landau–Lifshitz equation in tightly focused laser beams in the ultrarelativistic limit

The Landau–Lifshitz equation is widely considered as the equation, which correctly includes the effects of radiation reaction in the classical motion of an electric charge. Here, we provide an approximated analytical solution of the Landau–Lifshitz equation in the presence of a virtually arbitrary electromagnetic field by making the assumptions that the electron’s initial energy is the largest dynamical energy in the problem. We show that in a regime where radiation–reaction effects are a small correction of the Lorentz dynamics, the four-momentum of the electron can be determined perturbatively in the ultrarelativistic limit. We explicitly compute the electron four-momentum up to the first order and in the experimentally relevant case of an ultrashort, tightly focused laser beam as an external field