Depletion of Intense Fields

The interaction of charged particles and photons with intense electromagnetic fields gives rise to multiphoton Compton and Breit-Wheeler processes. These are usually described in the framework of the external field approximation, where the electromagnetic field is assumed to have infinite energy. However, the multiphoton nature of these processes implies the absorption of a significant number of photons, which scales as the external field amplitude cubed. As a result, the interaction of a highly charged electron bunch with an intense laser pulse can lead to significant depletion of the laser pulse energy, thus rendering the external field approximation invalid. We provide relevant estimates for this depletion and find it to become important in the interaction between fields of amplitude a0?103 and electron bunches with charges of the order of 10 nC