  The feasibility of complete photoionization experiments, in which the full
set of photoionization matrix elements are determined, using multiphoton
ionization schemes with polarization-shaped pulses has recently been
demonstrated [Hockett et. al., Phys. Rev. Lett. 112, 223001 (2014)]. Here we
extend on our previous work to discuss further details of the numerics and
analysis methodology utilised, and compare the results directly to new
tomographic photoelectron measurements, which provide a more sensitive test of
the validity of the results. In so doing we discuss in detail the physics of
the photoionziation process, and suggest various avenues and prospects for this
coherent multiplexing methodology.

Atomic Physics;Quantum Physics