Finite Rayleigh length in Compton backscattering

In this paper, an approach is proposed to take into account the influence of the finite Rayleigh length of a laser beam on the number of scattered photons in the process of inverse Compton scattering. The method is based on the calculation of the overlap integral between the particle distribution density functions in electron and photon beams, taking into account the change in their sizes and in a total cross section during the collision. It is shown that this effect can have a significant impact on the total number of scattered photons, reducing it. It occurs even at the Rayleigh lengths exceeding the dimensions of the interaction region. We estimated the effect for the parameters of LUXE facility and found it to be very weak. At the same time, for Gauss-Laguerre beams, in which due to the superoscillation effect the Rayleigh length seems to be able to reach very small values, we predict that the effect of the finite Rayleigh length in Compton backscattering will be very distinct, reaching about 20% or more.