  Frequency upshifts have been proposed as a first experimental signature of
collective effects in QED cascade generated electron-positron pair plasmas.
Since the high effective masses of generated pairs will reduce any frequency
change, stopped pairs at minimal Lorentz factor in the lab frame were thought
to be the dominant contribution to the the laser upshift. However, we
demonstrate that only considering stopped particles unduly neglects the
contributions of particles re-accelerated in the laser propagation direction.
Re-accelerated particles should, on a per particle basis, affect the laser more
strongly, and over a much longer timescale. To maximize particle contributions
to the laser upshift, we consider a Laguerre-Gaussian (LG) mode to better
reflect generated pairs. The LG mode doesn't have an advantage in particle
deceleration and re-acceleration when compared against a Gaussian beam, but the
LG mode can maintain particle contributions for a longer duration, allowing for
more pair density accumulation. Deceleration with a structured beam to keep
pairs within the laser should create a larger upshift, thereby lowering the
demands on the driving laser.
