  Electromagnetic showers developing from the collision of an ultra-intense
laser pulse with a beam of high-energy electrons or photons are investigated
under conditions relevant to future experiments on multi-petawatt laser
facilities. A semi-analytical model is derived that predicts the shower
multiplicity, i.e. the number of pairs produced per incident seed particle
(electron or gamma photon). The model is benchmarked against particle-in-cell
simulations and shown to be accurate over a wide range of seed particle
energies (100 MeV - 40 GeV), laser relativistic field strengths ($10 < a_0 <
1000$), and quantum parameter $\chi_0$ (ranging from 1 to 40). It is shown
that, for experiments expected in the next decade, only the first generations
of pairs contribute to the shower while multiplicities larger than unity are
predicted. Guidelines for forthcoming experiments are discussed considering
laser facilities such as Apollon and ELI Beamlines. The difference between
electron- and photon seeding and the influence of the laser pulse duration are
investigated.
