Scattering with total depletion in strong, focused fields

Theoretical approaches to QED scattering in strong fields typically treat the field as a fixed background with simple spacetime dependence, such as a plane wave. Two major challenges are therefore the inclusion of backreaction (e.g., depletion of the field) and spatial geometry (e.g., focusing). We show here that a solution to one problem can solve the other: even if particle wave functions in a chosen focused background are not known, we show they can be constructed once depletion is accounted for. We demonstrate this by giving the exact wave functions in a flying focus beam for which all energy is absorbed by particles scattering on it. In addition we use the wave functions to obtain a simple expression for the nonlinear Compton scattering amplitude, comparing with the plane wave case. Our methods thus open a new avenue of investigation in which two previously challenging effects are simultaneously brought under analytic control.