  Laser-electron beam collisions that aim to generate electron-positron pairs
require laser intensities $I \gtrsim 10^{21} ~\textrm{W/cm}^2$, which can be
obtained by focusing a 1-PW optical laser to a spot smaller than 10 $~\mu$m.
Spatial synchronization is a challenge, because of the Poynting instability
that can be a concern both for the interacting electron beam (if
laser-generated) and the scattering laser. One strategy to overcome this
problem is to use an electron beam coming from an accelerator (e.g. the planned
E-320 experiment at FACET-II). Even using a stable accelerator beam, the plane
wave approximation is too simplistic to describe the laser-electron scattering.
This work extends analytical scaling laws for pair production, previously
derived for the case of a plane wave and a short electron beam. We consider a
focused laser beam colliding with electron beams of different shapes and sizes.
The results take the spatial and temporal synchronization of the interaction
into account, can be extended to arbitrary beam shapes, and prescribe the
optimization strategies for near-future experiments.
